We first define the statistics of the small-scale velocity through what we call the ADSD. Then, we explicate how to sample the unresolved velocity from this ADSD.

In , the absolute diffusivity (i.e. KE times correlation time; ) of the unresolved velocity is twice the variance tensor trace, tr(a)=EσdBt2/dt, whereas the unresolved kinetic energy is tr(a)/dt. Clearly, this kinetic energy has no physical meaning. Indeed, it depends on the simulation time step, and one should have the possibility to choose the time step as close as possible to zero. Thus, the unresolved velocity amplitude is specified through an absolute diffusivity rather than through a KE. In the mathematics literature of homogenization, Kubo-type formulas may be seen as what physicists call absolute diffusivities. More generally, since the variance of a time-continuous white noise is infinite, it is more relevant to deal with absolute diffusivity rather than kinetic energy in order to describe the statistics of the time-uncorrelated velocity. Thus, keeping a spectral approach, we define – for any spatiotemporal field – an absolute diffusivity spectral density denoted A(κ) at the wavenumber κ. We will rely on this ADSD rather than on the KE spectrum, E(κ). Since the absolute diffusivity is the variance multiplied by the correlation time, it is naturally to define the ADSD as follows: 1A(κ)=△E(κ)τ(κ),whereτ(k)=1/κvκ=1/κκE(κ) is the eddy turnover time at the scale 1/κ and vκ is at characteristic velocity at this scale. Accordingly, we have 2A(κ)=κ-3/2E1/2(κ). If in addition we assume a KE self-similar distribution, 3E(κ)=C2κ-s, and we obtain 4A(κ)=Cκ-r, where r=(s+3)/2.

We aim at defining the unresolved velocity ADSD from the large-scale velocity. For this purpose, we will assume the self-similar model (Eq. ) is valid at all spatial scales. At each time step, we compute the ADSD of the large-scale velocity, Aw, from Eq. (). Then, we fit the coefficients C and r of Eq. (), as illustrated by Fig. . Let us denote these coefficients with Cw and rw. Note that they are time-dependent because they depend on w which is. More precisely, we estimate the coefficients Cw and rw in a wavenumber interval which approximately represents a inertial range of fully resolved scales (i.e. before the spectrum roll-off). In the left panel of Fig. , this interval is delimited by two vertical lines.

From there, we can define the unresolved velocity ADSD in such a way that the total velocity – resolved plus unresolved – meets Eq. () at small spatial scales with the same coefficients (Cw and rw). Since the two velocity components are not correlated, the total ADSD is the sum of the ADSD of each velocity component. In the inertial range, this reads 5A(κ)=Aw(κ)︸Leading atlarge scales+AσB˙(κ)︸Leading atsmall scales≈Cwκ-rw. Therefore, the unresolved ADSD is chosen to compensate the resolved ADSD roll-off – introduced by the deterministic subgrid parameterization – at small scales. Specifically, the unresolved ADSD is set to 6AσB˙(κ)=max⁡0,Cwκ-rw-Aw(κ)fBP2(κ). As previously, fBP2 is a band-pass filter between κm and κM. In practice, we set κM to the theoretical resolution, 7κM=πΔx, and κm to the effective resolution, which is estimated as follows: 8κm=ln⁡(0.95)ln⁡(0.1)1/pκM, where p is the order of the Laplacian used as a deterministic subgrid tensor (i.e. DbDt=-ν(-Δ)pb). The justification of the above formula is left in Appendix . Compared to the work of , the value of κm is less critical. Indeed, Eq. () implies a weaker unresolved ADSD at larger scales where the resolved ADSD, Aw, is stronger. This softens the threshold effect introduced by the band-pass filter fBP.

In practice, we set an upper bound for the estimation of rw. Without this upper bound, a concentration of energy at relatively large wavenumbers – scales smaller than κm – in the resolved fields can become unstable. Indeed, this localized energy concentration would decrease the rw estimation, and hence increase the unresolved ADSD AσB˙ through Eq. () at large wavenumbers – larger than κm. This implies a larger noise intake, which can induce a larger concentration of energy at relatively large wavenumbers in the resolved fields, resulting in a positive feedback loop. To prevent these unphysical instabilities, the slope rw is bounded.

We have proposed a way to compute the unresolved ADSD AσB˙ from the large-scale velocity statistics. From that ADSD, it is possible to sample the unresolved velocity, as explained in the following.

A 2D homogeneous divergence-free small-scale velocity can be constructed by filtering a 1D white noise B˙ with a 2D divergence-free filter σ˘=∇⊥ψ˘σ . 9σB˙=σ˘⋆B˙=∇⊥ψ˘σ⋆B˙, where ⋆ denotes a spatial convolution. This velocity field can be easily sampled in Fourier space. 10σB˙^(k)=ik⊥ψ˘σ^(k)B˙^(k)=1Δtik⊥ψ˘σ^‖k‖dBtΔt^(k), where dB^t is the spatial Fourier transform of dBt and dBtΔt is a discrete scalar white-noise process of unit variance in space and time. Thus, the small-scale velocity is defined by the Fourier transform of the streamfunction kernel, ψ˘σ^.

In order to link the unresolved ADSD to the kernel σ˘ which defines the unresolved velocity, we note that 11AσB˙(κ)=EσB˙(κ)dt=1μ(Ω)∮[0,2π]dθkκσ˘^(t,κ)2=2πκ3ψ˘σ^t,κ2, where μ(Ω) is the surface of the spatial domain Ω and θk is the angle of the wave vector k. From Eqs. ()–(), we can finally express the unresolved velocity as follows: 12σB˙^(t,k)=1Δtik⊥ψ˘σ^t,‖k‖dBtΔt^(k),13=1Δtik⊥max⁡(0,Cw‖k‖-rw-Aw(‖k‖))2π‖k‖3fBP(‖k‖)dBtΔt^(k).

Again the simulated unresolved velocity ADSD is physically relevant, while the KE spectrum is not. Indeed, the simulated unresolved velocity ADSD is expected to match the true (time-correlated) unresolved velocity ADSD, whereas the KE spectra of the simulated and true unresolved velocities differ. Indeed, the true unresolved velocity correlation time spectral distribution τ(κ) is not restricted to the time step dt.

The tuning-free Eq. () is attractive because it is not a priori restricted to stationary flows or to a particular type of 2D incompressible dynamics. In order to appreciate that, we can numerically compare it with the previous method of . For stationary, fully developed turbulence and after including a tuning stage to optimize the match, both parameterizations give approximately the same results. Therefore, in order to perceive improvements between the two simulations, we must either work with non-stationary flows or with erroneous tuning. This latter scenario is of course the meaningful one for the application of such models to the real world, where turbulence is non-stationary and heterogeneous, and so region-by-region and season-by-season tuning is impossible to do with quality checks in place.

We believe that non-stationarity may yield a fair, yet idealized, comparison. So, we here compare the two parameterizations in a non-stationary case. Figure  below shows simulated buoyancy fields initialized with “case 2” of and corresponding errors. After 2 d of advection, there is no turbulence yet, and a 128×128 resolution is sufficient to correctly resolve every scale. Therefore, no stochastic subgrid parameterization is needed. The self-similar method automatically adapts to the situation, whereas the method of introduces spurious buoyancy isoline roughness by randomly folding the isolines. Accordingly, the method of introduces more errors.

Our stochastic parameterization randomly folds tracer isolines. This process is often desirable. For instance, it can trigger physically relevant instabilities, such as the filament instabilities highlighted by . After these instabilities have been randomly triggered, eddies are formed by non-linear processes. In a similar way, Figs.  and  show that our stochastic dynamics enable a more-realistic eddy distribution than deterministic simulations. However, a homogeneous small-scale velocity may also perturb the tracer isolines which should remain still (i.e. which remain still in high-resolution deterministic simulations), e.g. sharp, straight, and coherent fronts. Figure  also highlights this drawback. A typical application of this problem in more realistic flow simulations is the simulation of jets like the Gulf Stream and regions of the Antarctic Circumpolar Current. These real-world jets are associated with diffusivity suppression , and this effect is not present in our formulation so far. If we seek to preferentially perturb some tracer gradients, a heterogeneous small-scale velocity is required. Note that the heterogeneity discussed here needs to be non-stationary and thus cannot be represented by the stationary EOF presented later in this paper. Besides, in a small ensemble of realizations, relevant heterogeneity of the small scales may make the spreading more accurate for UQ and enable comparable ensemble forecast accuracy with fewer members. We here propose a possible heterogeneous version of the previous method.

In order to obtain a heterogeneous model of the unresolved velocity, we need a heterogeneous version of the ADSD (Eq. ). Since the wavenumber, κ, cannot depend on the position, x, the constant, Cw, and/or the spectrum slope, r, should do. A spatially varying spectrum slope is probably difficult to estimate. Hence, we restrict ourselves to a spatially varying constant, Cw, and a spatially homogeneous spectrum slope. The constant may also varies with the time and the wavenumber. According to the Kolmogorov theory e.g. and Eq. (): 14Cw(x,t,κ)=cst.ϵFp(x,κ), where ϵF is the energy flux through the spatial scales and p=1/3 for a SQG flow (cst. is an abbreviation for constant). More specifically, the energy flux describes the energy moving from scales larger than 1/κ toward smaller scales and can be computed as follows: 15ϵF(x,t,κ)=△qκ<w⋅∇qκ<‾, where q is the transported (up to possible source terms) quantity, gκ< is the low-pass filtered version of g (setting to zero the Fourier modes of g which have frequencies larger than κ), and •‾ stands for the spatial average . For a SQG flow, q corresponds to the buoyancy normalized by the stratification: q=b/N. The energy flux is essentially a third-order moment. It is very important because it describes the cascade of the flow by non-linear energy transfers .

If the energy flux through scale is understood locally in space (as indeed , also assumes), Eq. () provides a natural parameterization of the unresolved velocity heterogeneities. We simply modulate the unresolved ADSD (Eq. ) by the heterogeneous ratio ϵFp/ϵFp‾, averaged over the resolved inertial range wavenumbers. 16σB˙̃^(t,k)=1Δtik⊥max⁡0,Cw‖k‖-rw-Aw‖k‖2π‖k‖3fBP(‖k‖)dBtΔt^k,17σB˙(t,x)=P{ϵFp(t,x)ϵFp‾(t)︸HeterogeneousmodulationσB˙̃(t,x)︸Homogeneousvelocity}, where P=Id-Δ-1∇∇T is the projector onto the space of free-divergence functions. This parameterization is physically meaningful, since, locally in space, a stronger direct cascade at large scales (larger ϵF and thus larger Cw) suggests that the unresolved velocity (large) should maintain this cascade by folding smaller-scale tracer structures. Furthermore, considering that the energy flux is a third-order structure makes this parameterization relevant to differentiate between strait fronts and curved structures (e.g. eddies). Indeed, at least three points are needed to define a curvature and differentiate between these structures. Figure  confirms that this modulation enables a more accurate spatial distribution of the stochastic folding.

In order to keep a divergence-free velocity and the ensuing properties (e.g. energy conservation), the modulated velocity is projected onto the space of free-divergence functions, using the operator P. Because of that we do not consider the advection correction w*-w=-12(∇⋅a)T of the LU formalism. Indeed, here the variance tensor has the simple form a=1dtr(a)Id. As such, the advection correction is a gradient field and is hence removed by the projection onto the space of free-divergence functions.

Although relevant, the comparison of Figs.  and  – about front dynamics – remains qualitative. For a quantitative demonstration, adapted metrics should be used. Indeed, simple isotropic, homogeneous second-order statistics (e.g. KE spectra) cannot distinguish between eddies and filaments. Bi-spectra may overcome this drawback, since they express three-point statistics. This quantitative analysis would necessitate studies of the metrics themselves. Therefore, these analyses will be addressed in future work.

Many other closures rely on the Kolmogorov model (Eqs. –): in particular, the famous Smagorinsky model and its variants e.g. provide a path to developing deterministic and dissipative scale-aware subgrid models. Typically, these models result in a Laplacian dissipation which involves a heterogeneous eddy diffusivity or viscosity coefficient, νSm. Aside from their heuristic theoretical justification, these Smagorinsky-type subgrid terms are formally equivalent to the turbulent dissipation of our stochastic model: ∇⋅12a∇q , where a=2νSmId and q is a transported quantity. This similarity suggests that a Smagorinsky-type model could provide a good estimate for our variance tensor and thus for the heterogeneity of the unresolved velocity.

The goal of the Smagorinsky model is to optimize the KE spectrum by targeting a specific turbulent diffusive scale adapted to the simulation resolution. A turbulent dissipation coefficient expression can be derived from the Kolmogorov model (Eqs.  and ) and the closure, 18ϵF=ϵD, where ϵD is the dissipation, a second-order moment related to the molecular or turbulent diffusion. To develop a Smagorinsky-type model, the resolved flux of the energy-like conserved invariant is equated to the dissipation of the energy invariant such that 19ϵD=νSm‖∇q‖2. From there, one can obtain an eddy diffusivity or viscosity coefficient proportional to ‖∇q‖h. For an SQG flow, the exponent is h=1/2, where q=b is the buoyancy .

In the Kolmogorov theory of homogeneous and stationary turbulence, the energy flux is a constant of the flow, and the closure (Eq. ) is an exact result of a simple energy budget over an ensemble mean. Indeed, there is no accumulation of energy at any scales in a stationary regime. This closure is very useful since the dissipation (Eq. ) is generally much simpler to compute than the energy flux. Nevertheless, in every flow realization, the energy flux and the diffusion vary with space and time, and they do not match each other locally. For instance, a strong straight front of an SQG flow involves a large dissipation but no energy cascade because the velocity is aligned with front (see Eq. ). Moreover, in any bounded, limited resolution situation, the inertial cascade range is limited so the energy flux through scale varies with the wavenumber, especially outside the inertial range.

The discrepancy between energy flux and dissipation is not so much of an issue for the Smagorinsky model because its aim is the optimization of a second-order statistics at small scales. Unfortunately, this closure cannot be used to simplify the modulation computation in our parametric random model. Indeed, the stochastic dynamics relies on processes – such as folding – associated with higher-order statistics. Figure  (middle right) illustrates this statement. The following unresolved velocity parameterization is used there: 20σB˙̃^(t,k)=1Δtik⊥max⁡0,Cw‖k‖-rw-Aw(‖k‖)2π‖k‖3fBP(‖k‖)dBtΔt^(k),21σB˙(t,x)=P{‖∇b‖1/4(t,x)‖∇b‖1/2‾(t)︸HeterogeneousmodulationσB˙̃(t,x)︸Homogeneousvelocity}. Along sharp straight fronts, the dissipation will be larger. Accordingly, the associated modulation ‖∇b‖1/4 enhances the stochastic folding where one would need it to weaken.

We introduce two types of velocity field:

a high-resolution velocity, v, on a fine mesh-grid (5122),

In order to estimate the EOFs, (ξi)i, involved in Eq. (), we assume that we can observe increments 26σΔBmΔT=σB(m+1)ΔT-BmΔT. We will interpret the following residual flow increments as a realization of the above: 27ΔX̃ijm=△Xij(mΔT,(m+1)ΔT)-X‾ij(mΔT,(m+1)ΔT).

The increments are supposed to be centred and divergence free (as ∇⋅ξi=0, ∀i). Therefore, after computing the residual flow increments, ΔX̃ijmm, they are centred, 28ΔX′ijm=△ΔX̃ijm-E^{ΔX̃ijm}, with the estimator 29E^{ΔX̃ijm}=△1N∑m=0N-1ΔX̃ijm, and projected onto the space of divergence-free functions, 30ΔXijm=△P{ΔX′ijm},withP=Id-Δ-1∇∇T.

Then, we can define the EOFs by an estimate of the spatial covariance of the residual flow increments (averaging over the time index, m). 31∑kξk(xij)ξkT(xpq)=σ(xij)σT(xpq)32≈1(N-1)ΔT∑m=0N-1ΔXijmΔXpqmT In order to properly define the EOFs, we must add the following orthogonal constraint: 33∫Ωξi(x)Tξj(x)dx=0ifi≠j.

Finally, after estimating the (ξk)k offline, the ensemble forecast can generated online with Eq. ().

This is a simplified model to describe the ocean surface dynamics at mesoscales (i.e. horizontal length scale of the order of 100 km) . The buoyancy, b, is transported at the ocean surface by a horizontal velocity field, 34DbDt=S, where DDt stands for the horizontal (deterministic or stochastic) material derivative and S represents possible sources and sinks. As the potential vorticity is assumed to be zero in the fluid interior, the Fourier transform of the velocity streamfunction, ψ^, is related to the Fourier transform of the buoyancy, b^, by the following SQG relationship: 35ψ^=1N‖k‖b^, where N is the stratification and k is the wave vector.

The LU and SALT versions of the SQG dynamics are formally similar to the deterministic model. However, the buoyancy transport (Eq. ) has to be understood in the stochastic sense (Eq. ). Furthermore, the SQG relationship (Eq. ) must be interpreted with 36w*=∇⊥ψ=∇⊥(-Δ)-1/2(b/N), in the SALT context, and w=∇⊥ψ=∇⊥(-Δ)-1/2(b/N), in the LU one. Besides, in the LU SQG, the advecting drift w⋆ has to be divergence-free. We enforce this constraint by projecting the drift correction (w*-w) onto the space of free-divergence functions. So, we have 37w*=Pw*-w+w=-12P∇⋅aT+∇⊥(-Δ)-1/2(b/N), where P=Id-Δ-1∇∇T. Indeed, the SQG model is derived from a QG model with a transport of the buoyancy at the surface. Then, the potential vorticity (PV) is assumed to be zero inside the fluid. In the SALT framework, the PV reads 0=PVslt=∇⊥⋅w*+f+f0/N2∂z2ψ, w*=∇⊥ψ and b=f0∂zψ. For the LU dynamics, 0=PVlu=∇⊥⋅w+f+f0/N2∂z2ψ, w=∇⊥ψ and b=f0∂zψ.

Nevertheless, the slight difference between the SQG SALT and the SQG LU models are not considered in this section since we neglect the advection correction of the LU framework (w*-w). So, we simulate the stochastic transport (Eq. ) coupled with the SQG relation (Eq. ). The unresolved velocity statistics encoded in σ are specified either by the self-similar method of Sect.  or by the data-driven method of Sect. .

We perform a high-resolution simulation (5122 spatial grid) of the deterministic SQG model with the following initial condition: 38b(x,t=0)=0.2B0cos⁡2πLx+y+Fx-x000+Fx-x100-Fx-x010-Fx-x110,39withF(x)=B0exp⁡-1215L2x2+y22,40andxij0=L411+L2ij. The source term, S, involves a hyperviscosity, a linear drag, and an additive stationary forcing such that 41S=△-νHVΔ4b-1τFb+AFsin⁡kFxxsin⁡kFyy. The parameters of the simulations are summed up in Table . The influence of the initial condition remains for about a month. Then, the turbulence is maintained by the forcing as illustrated by Fig. .

From t=50 d to t=100 d, the EOFs of the data-driven method are learned. From t=100 d to t=130 d, the deterministic high-resolution simulation is used as a reference. In this time interval, several stochastic low-resolution (642) simulations are performed and compared. These simulations are initialized at t=100 d with the reference high-resolution simulation projected at low resolution (i.e. keeping only the Fourier modes associated with a coarse-resolution grid).