1 A c omparison of stochastic and deterministic downscaling in eddy

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A comparison of stochastic and deterministic downscaling in eddy

resolving ocean modelling: the Lakshadweep Sea case study

Georgy I.Shapiro1, Jose M.Gonzalez-Ondina2, Mohammed Salim2, J.Tu2

1 University of Plymouth, Drake Circus, Plymouth PL4 8AA, UK

2 University of Plymouth Enterprise Ltd, New Cooperage Building, Royal William Yard, Plymouth,

UK, PL4 8AA P

Key words: ocean modelling, NEMO, downscaling, mesoscale, Indian Ocean

Key points:

 The paper compares two non-data assimilating regional models nested into the same data

assimilating global model.

 The first model (LD20_SDD) uses a stochastic-deterministic downscaling method while the

second model (LD20_NEMO) uses a dynamical downscaling.

 Both LD20 models show similar skills in reproducing temperature and salinity assessed against

observations in terms of root-mean-square of anomalies

 LD20_SDD has a better bias

 LD20_SDD is much faster and uses significantly less computational resources than

LD20_NEMO.

Corresponding author: Georgy I.Shapiro (g.shapiro@plymouth.ac.uk)

Abstract

This study compares the skills of two numerical models having the same horizontal resolution but

based on different principles in representing meso- and submesoscale features of ocean dynamics in

the Lakshadweep Sea (North Indian Ocean). The first model, titled LD20-NEMO, is based on solving

primitive equations using the NEMO (Nucleus for European Modelling of the Ocean) modelling

engine. The second one, titled LD20-SDD, uses a newer Stochastic-Deterministic Downscaling

method. Both models have 1/20o resolution and use the outputs from a Global Ocean Physics Analysis

and Forecast model at 1/12o resolution available from Copernicus Marine Service (CMEMS). The

LD20-NEMO uses only a 2D set of data from CMEMS as lateral boundary conditions. The LD20-

SDD consumes the full 3D set of data from CMEMS and exploits the stochastic properties of these

data to generate the downscaled field variables at higher resolution than the parent model. The skills

of the three models, CMEMS, LD20-NEMO and LD20-SDD are assessed against remotely sensed

and in-situ observations for the four-year period 2015-2018. All models show similar skills in

reproducing temperature and salinity, however the SDD version performs slightly better than the

NEMO version. This difference in resolution is particularly significant in simulation of vorticity and

computation of the share of the sea occupied by highly non-linear processes. While the NEMO and

SDD model show similar skill, the SDD model is more computationally efficient than the NEMO

model by a large margin.

Introduction

There is a growing tendency to move to higher and higher resolution in ocean modelling. Higher

resolution models are particularly helpful in simulations of ocean circulation in coastal and shelf seas

and in the vicinity of intensive jet currents such as the Gulf Stream or Kuroshio (Volkov et al. 2015;

Kang and Curchitser 2013; Kerry et al. 2016). The enhanced ability of a model to resolve mesoscale

and submesoscale eddies leads to significant improvement in simulation of large scale features such

as the Gulf Stream recirculation (Chassignet and Xu 2017). High resolution physical models provide a

solid background for the study and prediction of ecosystem dynamics and the distribution and

productivity of key marine species with remarkable detail and realism. Such ocean models underpin

sustainable resource management, improvement in food security and development of Blue Economies

(Solstice 2021). However, higher resolution comes at a cost. It is commonly accepted that the

increase of horizontal resolution in ocean models is associated with significant increase in required

computing power, typically by a factor of ten for each increase of the horizontal resolution by a factor

of two (Chassignet and Xu 2021). The enhancement of resolution by a factor of three from ORCA025

(1/4°) to ORCA12 (1/12°) grid in a global ocean model resulted in the 24-fold increase in

computational time on the UK Met Office supercomputer (Hewitt et al. 2021).

Therefore, a development of new time saving algorithms could provide a cost-effective solution in

high-resolution modelling. One of such algorithms titled Stochastic-Deterministic Downscaling

(SDD) was proposed in (Shapiro et al. 2021). It is based on the philosophy that at smaller scales the

ocean processes become more chaotic and resemble to some extent the dynamics of small-scale

turbulence which is studied by methods of statistical fluid dynamics (Monin and Yaglom 2013).

Hence, there is an intention of simulating small-scale ocean processes employing their stochastic

properties inferred from data in addition to deterministic properties inferred from equations of motion.

As a source of data the SDD method uses outputs from a coarser resolution (parent) ocean model. In

this study we contrast and compare the efficiency and accuracy of the SDD method against traditional

deterministic ocean circulation model in the Lakshadweep Sea located in tropical Indian Ocean.

Materials and methods

The area of study is located within 68E to 78E and 7.50N to 14.50N to the west of Indian peninsula

around the Lakshadweep archipelago containing 36 islands, atolls and coral reefs. The parent model is

the operational global model at 1/12 degree of resolution and 50 vertical layers available from

Copernicus Marine Service, product GLOBAL_REANALYSIS_PHY_001_030 (CMEMS, 2020).

This product is not available from CMEMS anymore and has been upgraded to product

GLOBAL_MULTIYEAR_PHY_001_030. The parent model assimilates observational data on Sea

Surface Temperature (SST), Sea Surface Height, and in-situ Temperature/Salinity profiles. The parent

model provides outputs, amongst others, of potential temperature, salinity, meridional and zonal

components of velocity. The model outputs are compared to three observational data sets: OSTIA

(2022), Argo float temperature/salinity profiles (Argo 2022) and GHRSST Multiscale Ultrahigh

Resolution (MUR) L4 analysis (GHR-MUR 2022).

LD20-SDD model

The child SDD-LD20 model has the same geographical limits as the parent model, however different

depth levels, also 50 in number, were selected to be better suited to the dynamics of the Lakshadweep

Sea than the parent model. The daily averaged outputs of temperature, salinity and horizontal velocity

are obtained by Statistical-Deterministic Downscaling from the parent to the child model. The SDD

method (Shapiro et al 2021) is based on the modified version of Objective Analysis (Gandin 1965;

Kalnay 2003) which is applied to the parent model output in order to downscale it to a finer (child)

model grid. The method treats fluctuations of field variables around their statistical means as a

random process to which Markov-Gauss algorithm can be applied in order to minimise, in statistical

sense, the error of calculation of field variables on the fine grid. The SDD method assumes isotropy

and local spatial homogeneity of the first and second statistical moments of the probability

distribution function. Local spatial homogeneity is defined in this case as small relative variations of

statistical moments over the length of one grid cell. The method allows to reveal details of oceanic

features which are only embryonically represented by the parent model. The SDD method requires

knowledge of the correlation functions of fluctuations of field variables. It uses the usual ergodic

hypothesis that replaces ensemble averages with time averages (Moore 2015). The slowly changing

averages are calculated using a moving time window. The length of the window is chosen to be long

enough to have sufficient number of members for averaging but short enough so that the seasonal

variability can be ignored.

In this study we used 11 days as the length of the time window and the total time period was two

years (01-01-2016 to 31/12/2017). The correlation function is calculated for each field variable Q and

for each parent grid node. First, the time averages E(

) within the time window centred at time

 are computed for all the nodes  = 1,2… , where  is the total number of nodes in the parent

mesh. The subscript w indicates that time averaging is done only within the temporal window. Then

fluctuations 

 = 

E

 are calculated at all grid nodes for the time point .

Fluctuations related to the same time point  but different nodes are used to calculate the products of

fluctuations 

 

, where 0 is the node under consideration or ‘central’ node. The process is

repeated for different ‘central’ nodes in the 3D parent model domain. Second, the time point  (and

the related moving average time window) are shifted by one time point (in our case one day) to

calculate the next set of averages, fluctuations, and their products. Third, the spatial correlations

Cor(,



) are computed between each ‘central’ node 0 and other grid nodes  at the same depth

level using the equation

Cor(

,

) = E(



)

std

 std(

) (1)

where E and std denote averaging and standard deviation respectively which are calculated over a

large time period (in our case two years). The process was repeated for different ‘central’ nodes.

The field variables are correlated through a number of process having different length scales.

Following the approach suggested in (Mirouze et al. 2016) we introduce two correlation length scales

—  and  for short-range and a long-range correlations respectively. They are estimated by

fitting, at every node, an isotropic Gaussian curve of parameters  ∈ [0,1] ,> 0 to the

correlation values obtained by Eq. (1).

′(, )= ()exp

()+(1 ())exp

() (2)

where  is the vector of coordinates of the node n0, and  is the distance between the nodes 0 and

. For eddy-resolving modelling we are interested in the short-range correlation represented by the

correlation length () , therefore in calculation of correlations using Eq. (1) we only include the

nodes  which belong to the ‘search area’ around each ‘central’ node, in this case it was 1.7×1.7

degrees in size (4–5 times greater than the anticipated short length scale). Once the correlations are

computed, only the short-length component of the correlation given in Eq. (3) are used for the

downscaling

(, )= exp

(), (3)

The computations according to Eq. (1) are carried out for a 3D array of ‘central’ nodes on the parent

grid to create a 3D array of correlation lengths. To give a feeling of the numbers, the total number of

n0 nodes of the parent model within the LD20_SDD domain is 306,106. As expected, the values of

() depend only weakly on  supporting the assumption of local statistical homogeneity.

Therefore, if  is the vector of coordinates of a node on the child rather than parent model grid, then

the correlation length () can be approximated by its value at neighbouring points. With this in

mind, and to reduce the computation times and the effect of outliers, we compute the fitting using Eq.

(2) for only every other node in each horizontal dimension of the parent mesh, while the correlation

lengths for other nodes are obtained by linear interpolation. The correlations thus computed are

smoothed layer-by-layer with a 2D Gaussian filter. This filtering respects the assumption of local

statistical homogeneity as the correlation lengths vary smoothly in space (Weaver and Mirouze,

2013).

Figure 1a and Figure 1b show examples of correlation data sets Cor(

,

) for SST calculated

using Eq. (1) at two different locations of the ‘central’ node 0 and the fitted Gaussian curves. For

comparison, Gaussian curves corresponding to the short length scales are superimposed. The

scattering of correlation coefficients is relatively small within the short (mesoscale) length and

become larger at greater distances. Similar graphs (not shown) were obtained for other field variable

and other depths levels. The smaller scatter at shorter distances is consistent with greater coherency of

ocean structures within meso and sub-mesoscale ranges.

a b

Fig. 1 Correlation data sets for SST at two locations of the ‘central’ node n0: (a) 10.7°N, 70.3°E and

(b) 7.3°N and 76.2°E. Blue dots represent correlation coefficients of SST fluctuations between the

central node and surrounding nodes . The solid red line represents the fitted two-scale correlation

function according to Eq. (2). The dashed line shows a superimposed Gaussian curve corresponding

to the short correlation scale in such a way that it can be visually compared to the two-scale one

According to Eq. (2), the correlation function is in general different for different ‘central’ node 0.

Figures 2(a-h) show spatial distribution of the correlation lengths across the domain at the surface and

at a depth of 156 m for temperature (T), salinity (S), U- and V- component of current velocity. Similar

maps are obtained for other depth levels of the parent mesh.

a b

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1Acomparisonofstochasticanddeterministicdownscalingineddyresolvingoceanmodelling:theLakshadweepSeacasestudyGeorgyI.Shapiro1,JoseM.Gonzalez-Ondina2,MohammedSalim2,J.Tu21UniversityofPlymouth,DrakeCircus,PlymouthPL48AA,UK2UniversityofPlymouthEnterpriseLtd,NewCooperageBuilding,RoyalWilliamYard,Plymouth,...

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