The aim of the present work is to numerically simulate observed sea surface currents with slow dynamics that is governed by the slow components of the forcings and with fast variations that parameterize the remaining, unresolved processes. This is achieved when the time-averaged dynamics and the statistics of the time increments of the model agree with the observation. In the following, the time series prefix “δ” gives the increment time series, δu(t)=u(t+δ)-u(t), and we call the observed fat-tailed PDF of the sea surface current increments, from , the “Exp-Lin” PDF, as it is composed of an exponential and a linear term.

When modelling a natural process, such as currents, through an SDE, part of the dynamics is resolved, and part of it is parameterized by noise. What is deterministic and what is noise depends on the degree of coarse-graining. In our previous work , the dynamics was parameterized, and the model has no predictability. The other extreme is a purely deterministic model that includes all the processes involved. Such model is unattainable. We implemented a hierarchy of three idealized local models (two velocity components at one point in space, evolving in time) for the sea surface current in the Gulf of Trieste. The first is the purely deterministic (DET) model, taking into account the tidal signal and the wind-forced Ekman dynamics, while the unresolved processes are neglected. The second is the Gaussian (GAU) model, which adds Gaussian additive stochastic noise for which the corresponding SDE can be solved analytically. The third is the superstatistical (SUP) model, which considers a deterministic signal with the most realistic stochastic noise. The stochastic part of this model is partially resolvable analytically. Furthermore, these models are forced by nine different forcing protocols (FPs) (see Sect. ). The deterministic and stochastic models come in pairs; the more processes and timescales are resolved deterministically, the less have to be parameterized stochastically.

More precisely, the total sea surface current vector uo is given in terms of the tidal current uM, the wind-forced Ekman sea surface current uE and the stochastic current uS, as detailed in Table for the different models. The tidal signal is an analytic and linear combination of n tidal components uM(t)=∑i=1nau,icos⁡(ωit-ϕu,i) (analogously for vM), where a, ω and ϕ are the amplitude, the frequency and the phase of each tidal component, respectively.

The Ekman current, subject to the wind forcing, the Coriolis force and the friction with the underlying water masses, is modelled by 1ddtuEvE=-CBh̃ũf-f-CBh̃ũuEvE+1h̃FuFv. In Eq. (), CB=ρocb, where cb is the underlying ocean layer drag coefficient; h̃=ρoh, where ρo is the ocean density and h is the considered ocean surface layer depth; and ũ is the total speed of the surface layer as defined in Table . The last does not include the tidal current, as it is assumed that the tidal signal is constant along the water column and does not create shear, while the Ekman and stochastic signals are present in the surface layer only. Finally, f is the Coriolis parameter, and F=ρaca|ua-uo|(ua-uo) is the wind forcing (ρa is the atmosphere density, ca the atmosphere drag coefficient and ua the wind velocity at 10 m above the sea surface). Equation () is an ordinary differential equation of the DET model, while it includes the stochastic quantity ũ in the GAU and SUP models. If the wind forcing were to cease, the modelled mean flow would fall to the tidal signal.

The Ekman modelization in Eq. () is valid only on frequency scales comparable to the Coriolis parameter. The stochastic velocity, representing the fast unresolved turbulent dynamics and present in the GAU and SUP models, is defined by the following set of equations (for the purpose of this study, i.e. to reproduce the observed superstatistical statistics, it is sufficient to have uncorrelated stochastic velocity components): 2dxy=-γxxydt+Qu00QvdWxdWy, 3ddtuSvS=-γuuSvS+ηxy. In Eqs. () and (), the variables dWx and dWy are derived from independent Wiener stochastic processes. For every independent realization of the Wiener processes Wx and Wy, we obtain a solution of the dynamics, named a random walker. The collection of all the random walkers forms the stochastic ensemble and allows construction of a PDF. The coefficient γx=1/τ is the inverse of the characteristic short timescale τ (short compared to the long timescale discussed below in connection with the SUP model), whose numerical value is determined from the observations , γu is a coefficient that is a posteriori adjusted to obtain the observed HFR initial decay of the autocorrelation function of the SUP model sea surface currents (Fig. 7) and η is a proper constant, as discussed below. The difference between the GAU and SUP models lies in the terms Q. In the GAU model, which is a simplification of the SUP model, Qu and Qv are constants: they are the mean values of the SUP model Q variables, as given in Table . The GAU model is a system of linear SDEs, and all variables are Gaussian and are determined by their means and variances. All the dependencies for means and variances on the original parameters are given in Appendix . Furthermore, in Eq. (), the analytical condition on η for which the Gaussian variable δuS has the same mean and variance as x (analogously δvS with y), given a fixed γu, is shown. This also helps to adjust the parameters of the SUP model.

In the SUP model, the Q terms originate themselves from 2ν stochastic processes αi: Q=∑i=02ναi2. The variable ν gives the degrees of freedom (DOFs), according to the interpretation of : the system has 1 DOF if its positive variable characterizing the variability of the system maximizes the entropy, i.e. it is exponentially distributed. This corresponds to half the value of the interpretation by . Each stochastic process αi is governed by 4dαi=-μαidt+βdWi,i=1,⋯,2νandβ=βufor theucomponentβvfor thevcomponent, where the variables dWi are derived from independent Wiener stochastic processes, βu and βv are constants whose values are determined empirically, and μ=1/T is the inverse of the characteristic long timescale T (i.e. τ≪T), corresponding to the variation timescale of the variance in the superstatistical approach. Its numerical value is determined from the observations . The variables αi are solutions of the Langevin equation: they are Ornstein–Uhlenbeck processes (we refer the reader to for a pedagogical discussion of stochastic processes), characterized by Gaussian statistics: 5pα(αi)=μπβ2e-μαi2/β2. In the SUP model, the variables Q depend on the variables ν and αi, as shown in Table . The Q variables are distributed according to a gamma distribution with a shape parameter equal to ν, given in Eq. () and Table . In Eq. (), we give the computation for its first moment E[Q], used in the GAU model. In Appendix , the PDFs of the variable x (and y) are derived for ν=2 and ν=1/2. A decrease in the stochastic degrees of freedom ν leads to a reduction of the variability (non-maximal entropy) in the unresolved processes, represented by the stochastic variables. In the case of 2 DOF, the variables x and y are Exp-Lin distributed with a PDF given by 6pν=2(x)=μγx2βue-2μγx|x|/βu2μγxβu|x|+1pν=2(y)=μγx2βve-2μγx|y|/βv2μγxβv|y|+1. In the case of 1/2 DOF, the variables x and y are distributed according to a modified Bessel function of the second kind of zero-order K0(|z|): 7pν=12(x)=2μγxπβuK02μγxβu|x|pν=12(y)=2μγxπβvK02μγxβv|y|. The SDEs for the variables uS and vS in Eq. () contain a linear drag term and are forced by a coloured non-Gaussian noise. We do not know an analytical distribution of uS and vS. Nevertheless, thanks to the definition of the constant η in Eq. () discussed in connection with the GAU model, the distributions of δuS and δvS are numerically similar to the PDFs of x and y (shown in Sect. ).

As it can be seen in Eqs. () and (), the final stochastic PDFs depend on the β parameter, which is linked to the variable's second-order moment, by E[x2]=β2μγx for ν=2. For this case, it will be shown in Sect.  that the PDF of the total velocity increment is given mainly by the stochastic velocity increment PDF only. For this reason, in the numerical simulations with 2 DOF in the stochastic part of the model, the value of β is therefore fixed by the observed variance s2 of the HFR velocity increment found in : β=sμγx. In the case of 1/2 DOF in the stochastic signal, the tidal and Ekman signals contribute more to the variance of the total velocity increment, but their contribution is unknown analytically. For this reason, in this case, the β coefficient is increased empirically in order to fit the observed variance in the total velocity increment PDF.

In Table , the definitions of the variables distinguishing the DET, GAU and SUP models are given with a summary of the main properties of the models' variables. For all the models, we can (or cannot) consider the eddy depletion term in the wind stress term, i.e. the relative velocity of the wind with respect to the sea surface current, causing a reduction of kinetic energy injection into the ocean , in contrast to the absolute wind velocity. We report the results just with the eddy depletion term, as it is the theoretically most correct formulation and does not significantly burden the computational calculation.

In summary, we developed three types of models: the DET model is purely deterministic with tidal and Ekman currents, the GAU model adds a Gaussian stochastic signal, and the SUP model considers a stochastic sea surface current with fat-tailed increments instead of the Gaussian noise (Table ).

The FRR relates the response of a system perturbed by an external perturbation to internal fluctuations of the unperturbed system. Exploring the FRR is particularly challenging and interesting when considering natural systems where perturbation experiments cannot be performed and statistical ensembles are not available. Our SUP model is an example of a subcomponent of the climate system, and it is instructive to explore the FRR for such model. We refer to for the theoretical background concerning FRRs. We show here some fundamentals of the theoretical concepts and the methodology we applied to the most involved of the SUP models (FP9, Table , Sect. ).

In the present case, given the SUP model evolutionary system with the SUP variables uo=uE+uM+uS and vo=vE+vM+vS, a small fixed perturbation ΔuS or ΔvS is imposed at t=t0 to each walker. Then, the system is left free to evolve and, for each time, the separation Δuo(t|t0)=uo,P(t)-uo(t) and Δvo(t|t0)=vo,P(t)-vo(t) between the perturbed (uo,P and vo,P) and the unperturbed (uo and vo) systems is computed. We emphasize that the methodology requires one velocity component to be perturbed and not both in the same perturbed simulation. It is possible to obtain the mean value of the perturbed variables through a mean response function Ruu(t) or Rvv(t): 8〈Δuo(t)〉=Ruu(t)ΔuSwhen uS is perturbed〈Δvo(t)〉=Rvv(t)ΔvSwhen vS is perturbed, where the angle brackets 〈⋯〉 indicate the ensemble mean. The FRR is concerned with expressing the mean response function in terms of correlation functions of the unperturbed system.

In practice, the mean response functions are computed as follows. After the first perturbation at t0, the selected variable in the perturbed system is perturbed again with the same variation at tk=t0+kΔt, and the separation between the perturbed and unperturbed systems is computed. The perturbation time is fixed to Δt=12 h, allowing us to repeat the procedure M=1311≫1 times, and ΔuS=ΔvS=8 cm s⁻¹. The diagonal mean response functions are computed as follows: 9Ruu(t)=1M∑k=0M-1〈Δuo(tk+t|tk)〉ΔuSwhen uS is perturbedRvv(t)=1M∑k=0M-1〈Δvo(tk+t|tk)〉ΔvSwhen vS is perturbed. According to , the FRR holds if the mean response functions have a connection with some suitable correlation functions computed in the unperturbed system. In particular, in the case of multivariate Gaussian variables, the mean response functions are a linear combination of the variable correlations. In Sect. , the results with unperturbed initial condition uo(t=0)=0 and when the stochastic signal is initially perturbed, as described before, is shown. Some brief comments for the case where the Ekman system is initially perturbed are also provided.

In order to test the predictability capabilities of the SUP model, perturbation methods are adopted assuming that the HFR observations are not affected by observational uncertainty and represent the reality, i.e. modelled data are replaced by observations. In the following, this method is called “observation-based perturbation”. In detail, at every time Δt, the perturbed system (uo,P) is updated to the observed HFR velocities (both the uo and vo components in the same simulation); in particular, it is the stochastic signal to be perturbed. The method may appear equivalent to the FRR method, but the equivalence is not valid because (i) the initial perturbation is not a constant but changes for each Δt time window and for each walker and (ii) we are perturbing both the uo and vo components (in the stochastic signal) in the same perturbed simulation. We define the following functions: 10ξ(t)=1M∑k=0M-1〈uo,P(tk+t)-uo(tk+t)2〉1M∑k=0M-1〈uo,P(tk-1+Δt)-uHFR(tk)2〉,11ϵ(t)=1M∑k=0M-1〈uo,P(tk+t)-uHFR(tk+t)2〉1M∑k=0M-1〈uo,P(tk-1+Δt)-uHFR(tk)2〉, where the time tk-1+Δt means tk right before the perturbation. The function ξ(t) quantifies the difference between the perturbed and the unperturbed system, while the function ϵ(t) computes the difference between the perturbed system and the observations. Both are normalized through the mean initial perturbation over all the time windows. In Sect. , the results are presented. Some brief comments for the case where the Ekman system is initially perturbed are also provided.

All the models start from the initial condition uo(t=0)=uHFR(t=0), where uHFR is the observed HFR sea surface velocity, with uS(t=0)=0. The obtained numerical sea surface currents are then compared to the HFR measurements. We start this section by commenting on FP1, which has no tides and no wind forcing (see Table ), and compare the DET, GAU and SUP models in the FP5 and FP9 cases (see Table ). We conclude this section with a discussion of the change in the stochastic DOF when the tidal forcing is modified. We show only time series for the u component of the sea surface current variables; the results for the v component are similar.

The stochastic model has to be adapted to the FPs to best represent the dynamics unresolved by the deterministic part of the model. FP1 does not take into account any physical forcings, and the deterministic part vanishes. The DET model is therefore null, while the GAU and SUP models have a nonzero stochastic part. The SUP model best fits the observed data when ν=2 DOF is chosen (see ), as we obtain a total velocity increment (δuo=δuS because, in FP1, δuM=δuE=0) distributed according to the Exp-Lin PDF (as summarized in Table ). The result is shown in the left column of Fig. , in which SUP δuo of FP1 coincides with SUP δuS of FP5 (as shown in Table  and commented on below, they both have ν=2 stochastic DOF). The GAU model in FP1 (Eq. () and the left column of Fig. , in which GAU δuo of FP1 coincides with GAU δuS of FP5) fails in reproducing the fat-tailed Exp-Lin PDF.

In FP5, a 12 h moving-averaged tidal and wind forcing is applied to the models, while FP9 includes the forcings without averaging (see Table ). For the different types of forcings, we choose the DOF that best fitted the superstatistical Exp-Lin PDF from . In particular, for FP5, ν=2 gives the best fits, while for FP9, ν=1/2 does. In Fig. , the PDFs of the stochastic variables of the GAU and SUP models for FP5 and FP9 are compared to the analytical PDFs. The figures validate the choice of the parameters in our model hierarchy for the different forcings, as summarized in Table : (i) the GAU δuS follows the analytical Gaussian from Eq. (); (ii) in FP5, the stochastic variable SUP x is Exp-Lin distributed, while in FP9, it is Bessel distributed; (iii) the SUP δuS are distributed as SUP x in both the simulations, showing the validity of the choice of η (fixed according to the analytical condition from the GAU model in Eq. () – its choice is discussed in Sect.  in connection with the GAU and SUP models). This also validates the numerical model through the cases where the analytical solution is known.

In Fig. , the time series of the observed and model velocities for some days of April 2021 are reported. The difference in the forcing timescale is clearly visible in the deterministic DET model velocities' variability: smoother in FP5 than in FP9. They both follow the slow variability of the observed HFR currents. In fact, looking at Fig. , the linear regression of the scattered data has a lower slope with respect to the bisector, meaning that the DET model is not able to simulate the observed extremes. Nevertheless, their 2D histograms have an elongated peak along the bisector, and their correlation coefficient reaches 55 % in FP5 and 56 % in FP9 for the u component (29 % and 35 % for the v component, respectively). Moreover, it is possible to see that sometimes (e.g. 3 April 2021 from Fig. ) the observed uHFR sea surface current shows a negative peak at the beginning of Bora events. This behaviour is missing from the DET model uo velocity. This fact can be due to the model instantaneous change in the values of h̃ and CB with the wind speed, while the sea surface system in nature takes some time to adjust to the wind forcing through vertical mixing (the thin surface layer suddenly accelerates, then thickens, and the speed decreases).

Regarding the ensemble averaged sea surface currents of the GAU and SUP models, they follow closely the DET model time series. This shows that the stochastic models increase the variability to observed values but have little influence on the averaged dynamics. We emphasize that if our model were perfect, the observations would be one of the walkers and therefore would at times be outside the SUP total velocity ensemble standard deviation σuo (red) band, as is the case for the two random walkers (green lines) reported in Fig. . The SUP ensemble standard deviation of the uE+uM velocity (light-blue band) is smaller than the SUP total velocity ensemble standard deviation σuo (red band), as expected, because the tidal velocity is purely deterministic and, in the Ekman velocity, the stochasticity enters only through the drag. What is particularly interesting is that the GAU σuo ensemble standard deviation (blue band) almost completely coincides with the SUP σuo (red band) and not with the GAU σuS (yellow band) ensemble standard deviation. A clear example of that is visible on 3, 4, 6 and 13 April 2021 in Fig.  for the u component, where the wind forcing increases and the GAU and SUP σuo standard deviations (blue and red bands, respectively) reduce with respect to the GAU σuS statistics (yellow band). In this case, it originates from the stochasticity present in the Ekman drag term.

Looking at the PDFs of the velocity increments in Fig. , it can be seen that the DET model PDF in FP5 is more peaked around zero compared to FP9, which is closer to the observations. This is because the latter model has a higher variability due to the inclusion of higher-frequency forcing data. Despite the rather high correlation of the DET currents with the observations, the DET model does not explain most of the variability seen in the HFR data, especially the tails, i.e. the extreme events, which are not well reproduced.

From Fig. , one can also observe that in FP5, the GAU δuo PDF is similar to the analytical PDF N(0,σδuS) from Eq. (), while in FP9, it deviates, showing a lower peak and fat tails. The reason is explained in more detail for the SUP model in the next paragraph, but, in summary, it is due to the capacity in FP9 of the tidal and Ekman modelization to resolve a large part of the variability. For both FPs, although the first and second moments of the velocities are comparable to the SUP model (Fig. ) and the shape of the PDF of the velocity increments has improved compared to the results of the DET model (Fig. ), the tails of the GAU velocity increment PDFs are (by construction) not fat enough to resemble the superstatistical Exp-Lin PDF, and the occurrence of extreme events is strongly underestimated.

Regarding the SUP model, in FP5, the SUP δuE+δuM PDF is very peaked, due to the 12 h forcing timescale that suppresses a large part of the variability. Hence, the SUP δuE+δuM PDF contributes little to the SUP δuo PDF, which is distributed similar to the SUP δuS Exp-Lin PDF, having fat tails (compare Figs.  and ). In FP9, the SUP δuE+δuM PDF is distributed closely to the DET δuo PDF (larger with respect to the FP5 case), meaning that a large part of the SUP δuo variability is resolved by the deterministic model. The SUP δuS PDF is given by the Bessel function of Eq. () and shown in Fig. . The SUP δuo PDF, shown in Fig. , is a convolution of the SUP δuE+δuM PDF (no analytical form available) and the Bessel function. This PDF shows a lower and shifted peak with respect to the observed and superstatistical Exp-Lin PDFs (probably due to the SUP δuE+δuM PDF), but its tails slightly deviate from the observed and Exp-Lin ones.

The percentages of the observed HFR velocity increments inside the SUP σδuo (and σδvo) ensemble standard deviation band are measured from the model results: 12FP5Prδuo-σδuo≤δuHFR≤δuo+σδuoSUPFP5≃73%Prδvo-σδvo≤δvHFR≤δvo+σδvoSUPFP5≃73%,13FP9Prδuo-σδuo≤δuHFR≤δuo+σδuoSUPFP9≃61%Prδvo-σδvo≤δvHFR≤δvo+σδvoSUPFP9≃62%. Repeating the calculation in Eq. () and Eq. () without the eddy depletion term, we obtain a difference of less than 1 % with respect to the shown case, in which the relative velocity between the sea and the atmosphere is considered. For FP5, the percentages compare well to the analytical superstatistical Exp-Lin values: 14Pr(-σ≤x≤σ)=∫-σ+σe-2|x|/σ2|x|σ+12σdx=∫0+σe-2x/σ2xσ+1σdx=12∫02e-z(z+1)dz≃73%. This fact quantitatively confirms that the SUP model in FP5 needs ν=2 DOF in the stochastic signal, producing a reliable PDF with respect to the observations. In this respect, the observations are therefore indistinguishable from a single walker of the SUP model ensemble. The percentages of the FP9 case, which has ν=1/2 DOF in the stochastic signal, show small discrepancies with respect to the analytical ones. Despite the deviations from the analytical superstatistical Exp-Lin PDF, the SUP model in FP9 reproduces the PDF tails of the velocity increments with even greater fidelity to the observations than in the FP5 case (Fig. ).

We now move our attention from the PDFs of the velocity increments to the PDFs of the velocities (Fig. ). The DET model is not able to represent the observed PDFs, with FP5 clearly missing the most of the variability and FP9 (full forcing) improving the situation but still not reproducing the fat tails. A stochastic part is needed to represent unresolved processes and to increase the variability. The GAU and SUP models can better reproduce the observations. The GAU model shows a good pattern around the peak, while it fails in the reproduction of the extreme events, with errors of around 1 and 3 orders of magnitude for the occurrence of extreme velocities of 0.5 m s⁻¹ in FP5 and FP9, respectively. The SUP model in FP5 confirms, also with this variable, that it can capture the observed statistics. In FP9, the SUP model shows a higher peak but demonstrates the ability to reproduce some details that FP5 cannot, as the PDF decays with two different slopes.

Regarding the deterministic DET model uo temporal autocorrelation functions shown in Fig. , the periodicity of the peaks are consistent with what is observed from the HFR data, while the amplitude of the peaks is higher with respect to the observations. This fact is mainly due to the high autocorrelation of the tidal signal and the absence of stochastic variability. The cross-correlation between the DET uM and the DET uE velocities shows a clear undamped periodicity of 1 d. This is due to the daily update of the forecast wind data set: it introduces a spurious artificial periodicity that perfectly correlates with the daily S1-component tidal variability. The observed temporal decay (with a correlation timescale of approximately 5 h) is well reproduced by the DET model uo autocorrelation from FP9, while it is overestimated by FP5. This can be easily explained: FP5 has a 12 h forcing timescale that brings spurious memory to the system.

The time decay and the modulation of the SUP uo and vo temporal autocorrelation functions are well reproduced in both simulations. The fact that the SUP initial decay is comparable with the HFR pattern shows that the value of the γu coefficient shown in Table  is well chosen. The amplitudes in FP5 underestimate the observed ones, while in FP9, the daily peaks are present. In both simulations, the SUP autocorrelation is more damped in time with respect to the DET one due to the introduction of the stochasticity, as expected. The long time variability of the correlation functions of the idealized models is not expected to be similar to the observed ones, as the model space domain is zero-dimensional and the equations do not include the dynamics of eddies and other coherent two-dimensional structures.

We have seen that when moving from longer-forcing-time-averaging FP5 to no-forcing-averaging FP9, the stochastic model needs fewer DOFs. In particular, the number of stochastic DOFs must decrease when the non-averaged tidal components are considered, as reported in Table . This can be seen numerically in Fig. , where all the PDFs are able to fit the observations appropriately. The rest of the FPs presented in Table  (FP3, FP4, FP6 and FP7) are not shown because of the following reasons. FP4 and FP7 do not take into account any wind forcing, so they are not able to mimic the observed velocities. FP3 and FP6, looking at the PDFs of SUP δuo and uo, give very similar results to FP5, with the main differences occurring in the height of the PDF peaks.

In order to check if the models capture the observed veering angle, the angle between the daily averaged wind and the daily averaged selected sea surface current is computed (a positive Ekman angle means that the sea surface current is on the right with respect to the wind). As seen in Fig. , the observed HFR Ekman angle shows a large spread for low wind regimes, while it accumulates around the observed mean value with stronger wind speeds. In both simulations, the Ekman angle obtained from the DET model (yellow line) collapses towards the same angle range for high wind speeds but with less variability with respect to the observations. The SUP mean Ekman angle is almost the same as for the DET model and converges towards the observed mean Ekman angle with increasing wind speed. The SUP modelled standard deviation (grey area) reflects the observed variability in the entire domain of the Ekman angle for low wind speeds and shows a decrease in variability for stronger wind speeds. The SUP modelled Ekman angle includes, most of the time, the observed Ekman angles inside the 1 standard deviation band. FP5 shows, for both the DET and SUP model, higher variability bands with respect to FP9 due to the higher number of stochastic DOFs.

In this section, the FRR is tested on the SUP model under the forcings of FP9. In Fig. , the perturbation methodology on the time series, described in Sect. , and the effects on the total sea surface velocity are visualized. In this figure, the uS velocity component is perturbed, while the independent vS component remains unperturbed. The total SUP uo velocity component is affected by a clearly visible perturbation that decays in time, while the total SUP vo velocity component is slightly perturbed through the Ekman drag term.

In Fig. , the diagonal response functions Ruu(t) and Rvv(t) with the correlations of the observed, unperturbed Ekman, stochastic, and total velocities and analytical functions discussed further are reported. The diagonal response functions show an exponential decay with timescale corresponding to the stochastic velocity drag coefficient γu (i.e. 1/γu≃140 min; see exponential fit in Fig. ). When it is uE or vE that is perturbed, the diagonal response functions show an exponential decay with timescale corresponding to the mean Ekman drag term CBũh̃ (i.e. h̃CBũ≃65 min; not shown). This reveals that any external perturbation is decaying due to the corresponding model drag (γu if we perturb the stochastic velocity; CBũh̃ if we perturb the Ekman velocity).

As expected, because uo and vo are not Gaussian variables and include terms that are not affected by the perturbation, as the tidal signal, the behaviour of the diagonal response functions differs from the corresponding autocorrelations. It is then not possible to describe the response functions as linear combinations of the correlations in the unperturbed system.

What is particularly interesting to observe in Fig.  is that the SUP uS and vS autocorrelations coincide with the analytical autocorrelation function for a stochastic Gaussian variable, defined through a linear SDE with coloured Gaussian noise, found by in his Appendix C2: 15CuS(t)=γue-γxt-γxe-γutγu-γx, showing a dependence on the drag coefficients only and not on the variance of the coloured Gaussian. This result seems surprising, as our stochastic velocities are obtained through a linear SDE with Bessel, not Gaussian, coloured noise. Our interpretation of the fact is the following: the stochastic velocity drag timescale 1/γu≃140 min is 1 order of magnitude smaller than the superstatistical Gaussianity timescale T≃2 d of the superstatistical analysis. This means that, under the correlation point of view, the stochastic velocity does not have the time to develop the non-Gaussian characteristics but operates at an almost constant variance. As a consequence, after a decay time of about 5 h, the SUP uS and vS autocorrelations collapse exponentially with the same γu coefficient of the diagonal response functions. This can also be checked analytically through Eq. () and knowing that γu<γx: 16CuS(t)=γxγu-γxγuγxe-γxt-e-γut⟶t→∞γxγx-γue-γut∝Ruu(t). Thus, knowing the autocorrelation function of the stochastic velocity, it is possible to obtain the response function of the total velocity. It is then possible to state that the FRR holds in the SUP model when the perturbation is applied to the stochastic signal.

In Fig. , the observation-based perturbation methodology on the time series, described in Sect. , and the effects on the total sea surface velocities are visualized. In this figure, the uS and vS velocity components are perturbed. Both the total SUP uo and vo show the observation-based perturbation subject to an evident decay in time.

The ξ(t) and ϵ(t) functions are reported in Fig.  and represent the normalized distance between the perturbed–unperturbed and the perturbed–observed systems, respectively. The function ξ(t) can be considered as a generalization of the response function in the FRR when both the components are perturbed of a quantity not known a priori. It is possible to see that the ξ(t) function has an exponential decay with timescale 1/γu≃140 min. When the Ekman signal is perturbed, we obtain two exponential decays: τ1≃75 min for t<2 h and τ2≃140 min for t>4 h (not shown). The external perturbation, due to the observation-based methodology, decays with the stochastic drag coefficient when we perturb the stochastic velocity, while a clear correspondence with the drag coefficient is not found when the Ekman signal is perturbed. The ϵ(t) function increases its value in time from 0 and, after about 5 h, saturates to 1, showing that, after that time, the perturbed system has completely lost the memory of the initialization to the observations. This time of 5 h coincides with the autocorrelation timescale of the SUP stochastic and total velocities.