An idealized study of coupled atmosphere­ocean 4D­Var in the presence of model error Article 

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Fowler, A. M. and Lawless, A. S. (2016) An idealized study of coupled atmosphere­ocean 4D­Var in the presence of model error. Monthly Weather Review, 144 (10). pp. 4007­4030. ISSN 0027­0644 doi: https://doi.org/10.1175/MWR­D­15­0420.1 Available at http://centaur.reading.ac.uk/65663/ 

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An idealized study of coupled atmosphere-ocean 4D-Var in the

presence of model error

Alison M. Fowler and Amos S. LawlessSchool of Mathematical and Physical Sciences, University of Reading, Reading, RG6 6AX, UK.

Abstract

Atmosphere only and ocean only variational dataassimilation (DA) schemes are able to use win-dow lengths that are optimal for the error growthrate, non-linearity and observation density of therespective systems. Typical window lengths are6-12 hours for the atmosphere and 2-10 days forthe ocean. However, in the implementation ofcoupled DA schemes it has been necessary tomatch the window length of the ocean to thatof the atmosphere, which may potentially sac-rifice the accuracy of the ocean analysis in or-der to provide a more balanced coupled state.This paper investigates how extending the win-dow length in the presence of model error affectsboth the analysis of the coupled state and theinitialized forecast when using coupled DA withdiffering degrees of coupling.

Results are illustrated using an idealized singlecolumn model of the coupled atmosphere-oceansystem. It is found that the analysis error froman uncoupled DA scheme can be smaller thanthat from a coupled analysis at the initial time,due to faster error growth in the coupled sys-tem. However, this does not necessarily lead toa more accurate forecast due to imbalances inthe coupled state. Instead coupled DA is moreable to update the initial state to reduce the im-

pact of the model error on the accuracy of theforecast. The effect of model error is potentiallymost detrimental in the weakly coupled formu-lation due to the inconsistency between the cou-pled model used in the outer loop and uncoupledmodels used in the inner loop.

1 Introduction

Coupling processes between the atmosphere andocean are known to be important for seasonaland climate prediction, for example for the ac-curate prediction of the El Nino-Southern Os-cillation (Barnett et al. 1993; Jin et al. 2008).In addition to this, there is increasing evidenceto suggest the importance of atmosphere-oceaninteraction at the weather time scales, for ex-ample for the prediction of the Madden-JullenOscillation (Shelly et al. 2014), coastal fog andextra-tropical cyclones (Siddorn et al. 2014; Vi-tart et al. 2008).

Until recently the initialization of coupledmodels for the prediction of the Earth system hasbeen performed using uncoupled atmosphere andocean states, produced from separate data as-similation (DA) systems (e.g. Saha et al. (2006);Molteni et al. (2011); Arribas et al. (2011);MacLachlan et al. (2015)). These uncoupledstates are then effectively ‘stitched’ together to

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create an initial state for the coupled forecast.There are a variety of known problems with us-ing uncoupled initial conditions. These include

• the generation of imbalances between theatmosphere and ocean systems leading to anunrealistic and sudden adjustment to bal-anced conditions in the first part of the fore-cast (Balmaseda and Anderson 2009; Mull-holland et al. 2015); and

• a sub-optimal use of observations, partic-ularly those close to the air-sea interfacewhich depend on the physics of both theocean and atmosphere.

In an attempt to overcome many of these issues,coupled DA methods, in which the atmosphereand ocean are treated simultaneously, are be-ing developed at operational centers worldwide(Saha et al. 2010; Lea et al. 2015; Alves et al.2014; Laloyaux et al. 2015). As well as for theinitialization of coupled models, coupled DA willbe used for reanalyses, in which it is necessaryto provide a consistent global transport of mass,water, and energy on the relevant time scales(Dee et al. 2014).

Coupled DA is a relatively new field of re-search. Early studies include those by Galantiet al. (2003), Zhang et al. (2007) and Sugiuraet al. (2008). Despite the techniques proposedin these early studies not estimating the full at-mosphere and ocean states simultaneously, re-sults showed that using a coupled atmosphere-ocean model during the assimilation significantlyimproved the accuracy of reanalyses and fore-casts of seasonal to interannual climate varia-tions (such as El Nino). Another early study isthat of Lu and Hsieh (1998), who used a simple5 dimensional coupled equatorial model to inves-tigate the use of coupled DA for both state and

parameter estimation. Within this study we fo-cus on the use of coupled DA for short-mediumrange weather forecasting which can be sensi-tive to the interaction between the atmosphericboundary layer and the oceanic mixed layer.

Implementing a fully coupled DA scheme inpractice faces many challenges. An immediatepractical consideration is the increase in the sizeof the state that needs to be estimated, whichmakes the problem much more computationallyexpensive. There are also more scientific chal-lenges which are invariably related to the verydifferent nature of the two fluids. The atmo-sphere is much less dense than the ocean andalso much more unstable. There are also funda-mental differences in the available observationsof the atmosphere and ocean. The atmosphere isrelatively densely observed whilst the ocean byits very nature is difficult to observe and sub-surface observations generally only come from asparse array of in-situ observations. These dif-ferences impact on essential aspects of the im-plementation of coupled DA, such as the specifi-cation of cross-covariances between errors in theatmosphere and ocean systems (Han et al. 2013).

In previous work, we used an idealized singlecolumn model to investigate different approxi-mations to coupled DA in the 4D-Var frameworkwhen no model error is present (Smith et al.2015). It was found that when using a shortwindow length of 12 hours (consistent with thatused operationally for atmosphere only DA), astrongly coupled formulation was able to pro-vide the best analysis of the coupled initial state,having both a smaller error and being more bal-anced than an uncoupled formulation. A weaklycoupled approximation was seen to display someof these same benefits as the strongly coupledformulation but was more sensitive to the reso-lution of the observations. In this paper we con-

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tinue this study, by investigating how the differ-ent approximations to coupled DA compare asthe window length is extended and model errorbecomes a greater issue.

Due to the different nature of the two fluids,model error in the atmosphere has a much fastergrowth rate than in the ocean. The growth rateof the model error and the linearity assumptionrestricts the length of the window in which obser-vations can be assimilated in 4D-Var, and is animportant factor in why a short window lengthof 6-12 hours is typically used in atmosphereonly data assimilation (Rawlins et al. 2007), anda long window length of 2-10 days is typicallyused in ocean only data assimilation (Weaveret al. 2003). Coupling the two systems allowsfor the model error to interact and may intro-duce a faster model error growth rate in theupper ocean than in an uncoupled simulation.Therefore the model error in the coupled systemrestricts the assimilation window to somethingshorter than the optimal window length for anuncoupled ocean DA scheme (Lea et al. 2015;Laloyaux et al. 2015). As the ocean is poorlyobserved, this severely limits the number of ob-servations that can be assimilated and has theeffect of potentially sacrificing the accuracy ofthe ocean initial state in order to provide a morebalanced coupled initial state. Even with theuse of a shortened window length, the ECMWFhave found it necessary to constrain the sea sur-face temperature to a gridded analysis productin order to avoid the rapidly growing bias in themodel (Laloyaux et al. 2015). There is thereforemotivation to understand the effect of extendingthe window length, in the presence of increasingmodel error, within the coupled DA framework.

The model error in the coupled system wasstudied by both Magnusson et al. (2013) andSmith et al. (2013) using the ECMWF IFS model

coupled to the NEMO ocean model (Madec2008) and the HadCM3 model respectively. Bothfound cold biases in surface temperatures. Mag-nusson et al. (2013) believed this is due to im-balances in the energy flux at the top of theatmosphere and a strong uptake of heat in theocean. Magnusson et al. (2013) also looked atthe bias in the 10m wind speed and found itto be large within the western Tropical Pacific.They concluded that this is due to a positive cou-pled feedback between wind and SST: too strongwinds lead to excessive upwelling which producesa colder sea surface temperatures (SST) which inturn produces stronger zonal winds.

Here we examine how model errors in the cou-pled system affect the analysis and subsequentforecast in coupled DA. The structure of the pa-per is as follows. In section 2 the different ap-proximations to coupled DA that we consider aregiven. This is largely an overview of results re-cently presented in Smith et al. (2015). In sec-tion 3 the theoretical impact of model error ispresented with discussion of how the differentcoupling strategies may be affected. In section4 the design of the idealized experiments is pre-sented and an illustration of the model error fora case study is shown. In section 5 results areshown for a series of assimilation experimentsapplied to this case study in which the effect ofmodel error on the accuracy of the analysis andinitialized forecast are given. Finally conclusionsthat can be drawn from these experiments aredetailed in section 6 along with a discussion ofthe insight provided on how to account for modelerror in coupled 4D-Var.

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2 Strategies for coupled DA

This present study will focus on incremental 4D-Var methods (Courtier et al. 1994) in line withthose being developed at ECMWF and the MetOffice (Laloyaux et al. 2015; Lea et al. 2015).We note that methods based on the ensembleKalman filter are also being developed with in-teresting results (e.g. Frolov et al. (2016) andZhang et al. (2007)) but will not be includedin the comparison of methods presented here.We consider three different strengths of couplingwithin the 4D-Var scheme:

1. strongly coupled, in which the ocean and at-mosphere are updated together. This is theepitome of what coupled DA schemes aretrying to achieve, but is currently not prac-tical for operational sized models;

2. weakly coupled, an approximation tostrongly coupled which makes use of theinner loop structure of incremental 4D-Varto simplify the algorithm. This makes itfeasible to implement in current operationalsettings, particularly when different assim-ilation schemes have been implemented inthe atmosphere and ocean;

3. and lastly uncoupled in which, the atmo-sphere and ocean are updated indepen-dently of each other.

An in-depth description and study of these dif-ferent coupling strategies in the case of no modelerror is given in Smith et al. (2015). Here a briefsummary of the different algorithms is given.

As stated above, each method is based on in-cremental 4D-Var which minimizes iteratively aseries of linear approximations to the full 4D-Var

cost function given by

J(x0) = 12(x0 − xb

0)TB−1(x0 − xb0)

+12(y − H(x0))

TR−1(y − H(x0)).(1)

The two sources of information about the initialstate are given by xb

0 , the background state (orfirst guess, usually provided by a previous fore-cast valid for the time of interest) and y, a vectorof all observations throughout the assimilationwindow. The operator H(x0) is the generalizedobservation operator, a mapping from the initialstate to all the observation variables and times.This differs from the observation operator, Hi(.),which maps the model state at time ti to obser-vation variables at time ti. B is the backgrounderror covariance matrix and R is the observationerror covariance matrix.

The incremental 4D-Var algorithm can be ex-pressed in terms of a series of outer and innerloops, illustrated in figure 1. In the outer loopthe linearization state, xg

0:n over the time window[t0, tn], and the innovations, δy, are evaluated.The linearization state is given by

xg0:n =

xg0

M(xg0, t0, t1)...

M(xg0, t0, tn)

, (2)

where n is the number of time steps in the assim-ilation window and M(x0, t0, ti) is the propaga-tion of the model state at time t0 to time ti. Thevector xg

0 refers to the current estimate of the ini-tial state; initially this will be the background,but it will then be updated on each outer loop.The innovations are then given by the differencebetween the observations and the estimate of theinitial state mapped to observation space,

δy = y − H(xg0). (3)

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The computation of the initial state mapped toobservation space, H(xg

0), utilizes the lineariza-tion state in the following way:(H(xg

0))i

= Hi(M(xg0, t0, ti)) = Hi(x

gi ). (4)

In the inner loop a linear approximation tothe full cost function (1) is then minimized withrespect to δx0 = x0 − xg

0:

J(δx0) = 12(δx0 − (xb

0 − xg0))TB−1(δx0 − (xb

0 − xg0))

+12(δy − Hδx0)

TR−1(δy − Hδx0)),(5)

where Hδx0 is a tangent linear (TL) approxi-mation to H(x0)−H(xg

0), linearized about xg0:n.

This can be separated out in terms of the TLapproximation to the observation operator, H,and the dynamical model, M.

H =

H0

H1M0:1...

HnM0:n

, (6)

where

M0:i =∂M(x0, t0, ti)

∂x0. (7)

Once the minimization of J(δx0) has met agiven criterion, the estimate of the initial stateis updated: xg

0 := xg0+δx0. The ‘analysis’, which

is used to initialize the forecast, is then given byxg0 once the outer loop has converged, or a max-

imum number of iterations has been performed(Courtier et al. 1994; Lawless 2013).

2.1 Strongly coupled

In strongly coupled DA the state vector, x0, in-cludes both the atmospheric and oceanic vari-ables. The non-linear (NL) model, M, used tocalculate (2) is the fully coupled model and the

exact TL approximation to H(x0) − H(xg0) is

used in (5). Within this formulation, the co-variance matrix B may include cross covariancesbetween errors in the atmosphere and ocean andthe observation operator may account for thesensitivity of observations to both the atmo-sphere and ocean. Exactly how to derive thebackground error cross covariances between theatmosphere and ocean is an area of active re-search (e.g. Han et al. (2013)).

A benefit of 4D-Var over 3D-Var is that theinclusion of the model dynamics allows for someflow-dependence to develop in the backgrounderror covariances (Thepaut et al. 1996). In Smithet al. (2015) strongly coupled DA was found tobe able to generate implicit correlations betweenthe atmosphere and ocean states and hence ob-servations of the atmosphere were able to influ-ence the analysis of the ocean and vice versa,even when no explicit correlations in the back-ground errors were specified. This provided amore balanced analysis and allowed for more in-formation to be extracted from the observations.

2.2 Weakly coupled

In weakly coupled DA the coupling between theatmosphere and ocean is only accounted for inthe outer loop. In practice, this means that thestate vector, x0, still includes both the atmo-spheric and oceanic variables and the NL coupledmodel is still used to calculate (2). However, in-stead of using the exact TL approximation asin strongly coupled DA, an uncoupled approxi-mation is used. Therefore (5) can be split intotwo cost functions, one corresponding to findingthe increment to the atmospheric state, δxatmos

0 ,and the other corresponding to finding the incre-ment to the oceanic state, δxocean

0 . This meansthat although it is possible for the observation

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Figure 1: Left: Schematic of the incremental 4D-Var algorithm. Right: Illustrationof the the non-linear cost function (blue) and the linear approximations made oneach inner loop (green).

operator to account for the sensitivity of obser-vations simultaneously to both the atmosphereand ocean in (3), the calculation of δxatmos

0 andδxocean

0 in the minimization of (5) does not ac-count for this. Similarly it is not possible to in-clude cross covariances between background er-rors in the atmosphere and ocean.

As suggested by the name, this formulation re-duces the strength of the coupling. Since the in-novations (3) are computed in observation spaceand H is able to contain contributions from boththe updated ocean and atmosphere, the strengthof the coupling can be seen to be related to thedensity of the observations, particularly thosewhich are sensitive to the coupling of the oceanand atmosphere. However as B, H and M arenot coupled, atmospheric observations cannotupdate the ocean state directly, and vice versa.Therefore, even if cross correlations between theatmosphere and background errors were not in-cluded in strongly coupled DA, it is not simplya case of the weakly coupled scheme converging

to the strongly coupled scheme as more outerloop iterations are performed. This means thatcompared to strongly coupled DA the risk of ini-tialization shock is increased and weakly coupledDA is particularly sensitive to the frequency anddensity of observations (Smith et al. 2015).

This approach is the essence of that currentlybeing investigated by the ECMWF (Laloyauxet al. 2015) and Met Office (Lea et al. 2015).However, in the initial implementation at thesecentres there are some differences to the cleanform we present here. Most notably, both theECMWF and UK Met Office only use 4D-Var forthe atmosphere and instead use 3D-Var FGAT(first guess at appropriate time) for the ocean.Another important point is that the UK MetOffice currently only perform one outer-loopwhilst the ECMWF perform two, increasing thestrength of the coupling.

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2.3 Uncoupled

In uncoupled DA the state vector is separatedbetween the atmosphere and ocean components.The uncoupled non-linear models are used tocompute (2), with the boundary conditions atthe interface specified externally, and the exactTL approximation of each uncoupled model isused in the minimization of (5) to approximatethe non-linear models used. There is thereforeno exchange in information between the atmo-sphere and ocean and the analysis incrementsδxatmos

0 and δxocean0 may be inconsistent. It is

this inconsistency that can lead to initializationshocks.

Our aim is to understand how each of thesecoupling strategies react to errors in the cou-pled model equations. In particular, we examinewhether there may be benefit to using the un-coupled and weakly coupled formulations if theerror growth rate is larger in the coupled modelcompared with the uncoupled model. In the nextsection a brief summary of the theory of modelerror in 4D-Var is given along with an exami-nation of how model error may enter into thedifferent coupling strategies.

3 Model error in 4D-Var

There are many different potential sources ofmodel error in coupled atmosphere-ocean mod-els. In practice it is rare to be able to identify(and correct) a single cause of model error withinthe model itself. Nevertheless attempts havebeen made. For example Vanniere et al. (2014)demonstrated a systematic approach for iden-tifying sources of error in tropical SSTs. Thisinvolved performing 7 separate simulations us-ing coupled and uncoupled models, with differ-

ent coupling, forcing and initialization strategies.Such a method shows promise for aiding modeldevelopment to reduce model error but it cannoteradicate it.

In 4D-Var, model error can be seen to manifestitself through the calculation of the innovations,(3), in which it is assumed that H(x0) providesthe exact mapping between state and observa-tion space. In the presence of model error thisno longer holds. Instead, under the assumptionof additive model error, we have the following re-lationship between the truth in state space, xt

0,and the truth in observation space, yt:

yt = Ht(xt0)

= H(xt0) + εH,

(8)

where Ht is the exact mapping but H is themapping used within the assimilation. The error

in the generalized observation operator, εH, hasthe same dimensions as the observation vectorand incorporates error in the model equationsdescribed in (2). Within this current work it isassumed that the error arises from the dynamicalequations rather than the observation operatorH. Errors may also be present in the observa-tion operator, which are often considered to beerrors of representativity. This is a vast error ofresearch (e.g. Waller et al. (2013), van Leeuwen(2014), Bormann et al. (2014)) but will not beconsidered further in this current work, that iswe will assume the observation operator, H, tobe perfect.

If model error is unaccounted for then B, Rand H remain unchanged in the 4D-Var algo-rithm. Therefore the computation of the analy-sis, xa, can still be given by the theoretical linearapproximation:

xa = xb + K(y − H(xb)), (9)

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where K is known as the Kalman gain matrixgiven by BHT(R + HBHT)−1. With multipleouter loops xb in (9) would be replaced by thecurrent outer loop iterate, and K computed witha H linearized around the current outer loop tra-jectory. However, as the generalized observationoperator is no longer optimal the error covari-ance of the analysis will be inflated, with covari-ance equal to

Pa+KE[(εH−E(εH))(εH−E(εH))T]KT, (10)

where E[.] is the mathematical expectation andPa is the analysis error covariance if H(xt

0) werecorrect. In deriving (10), it is assumed that themodel error is uncorrelated with the observationand background error. In addition to this, if the

random variable εH is biased then the analysiswill be biased, that is

E[εa] = KE[εH], (11)

where εa = xa − xt0 (the analysis minus the true

initial state). The derivations of (10) and (11)can be found in the appendix. In a similar way,it can be shown that the expected value of thecost function evaluated at the analysis will alsobe increased, as it becomes impossible to fit toboth the background and the observations in away which is consistent with the prescribed errorcovariances (Dee 1995).

From equations (10) and (11) we see that theimpact that the model error has on the analysiserror depends on the Kalman gain matrix.The larger the elements of K are, the greaterthe impact of model error, or in other wordsthe more dominant the observations are (dueto either their number or their accuracy) incomputing the analysis, the greater the impactof model error.

In addition to the non-linear model errorpresent in the outer loop, an error in the TLapproximation to the NL model used in the as-similation is also present in the inner-loop of in-cremental 4D-Var. Let

εTL = H(x0 + δx0)− H(x0)− Hδx0 (12)

be the tangent linear model error in observationspace. For each of the coupling strategies thechoices of H and H differ and so the error inthe generalized observation operator and the TLmodel error is different in each case:

Strongly coupled:

εH = Ht(xt0)− Hc(xt

0) (13a)

εTL = Hc(xg0 + δx0)− Hc(xg

0)− Hcδx0 (13b)

Weakly coupled:

εH = Ht(xt0)− Hc(xt

0) (14a)

εTL = Hc(xg0 + δx0)− Hc(xg

0)− Hucδx0 (14b)

Uncoupled:

εH = Ht(xt0)− Huc(xt

0)(15a)

εTL = Huc(xg0 + δx0)− Huc(xg

0)− Hucδx0(15b)

The superscripts c and uc refer to the cou-pled and uncoupled assimilation models. Wenote that the truth, Ht(xt

0), is always coupledby definition. From these equations it is clear

that εH is the same for the weakly and stronglycoupled formulations due to the outer-loop cal-culations being the same. For the strongly cou-pled and uncoupled methods the exact tangent

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linear of the erroneous nonlinear model is usedin the inner loop. In these cases the incremen-tal 4D-Var scheme is equivalent to a truncatedGauss-Newton iteration and, provided that theinner loop is solved to sufficient accuracy, theouter loop iterates should converge to a mini-mum of the corresponding discrete cost functionand the TL error should tend to zero. On theother hand, for the weakly coupled assimilationthe uncoupled tangent linear models only pro-vide an approximation to the true linearizationof the coupled model used in the outer loop. Inthis case the incremental 4D-Var scheme is a per-turbed Gauss-Newton method and, under cer-tain conditions, will converge to a solution closeto the minimum of the discrete nonlinear costfunction, but not equal to it (Lawless et al. 2005;Gratton et al. 2007). In practice, however, thetangent linear model always contains some ap-proximations, since it is usually run at a lowerresolution than the outer loop nonlinear modeland may not include the linearization of all phys-ical parameterizations.

There are also differences in the way the lin-earization trajectory is defined. For the weaklycoupled case, Huc is linearized about the cou-pled trajectory given by Hc(xg

0) and the bound-ary conditions at the air-sea interface (BCs) usedto force Huc are calculated using the NL cou-pled model. For the uncoupled case, Huc is lin-earized about Huc(xg

0). The BCs for the un-coupled model runs, Huc(xg

0) and Huc, are pre-scribed externally.

4 Experimental design

Following on from Smith et al. (2015) we makeuse of a single column model of the coupled at-mosphere and ocean. We aim to create an ex-

perimental set up in which we believe the modelerror to have characteristics of the error seen inan operational coupled atmosphere-ocean model.That is, we wish for the growth rates of themodel error in the atmosphere to be much largerthan in the ocean and to be complex in nature.Below details of the ‘true’ and erroneous modelare given. We assume the error to originate frommissing physics, erroneous parameter values anderrors in the large scale forcing.

4.1 ‘Truth’ model

The model which represents the truth in ouridealized experiments comprises of the ECMWFsingle-column model (SCM), which originatesfrom an early cycle of the IFS (Integrated Fore-casting System) code, coupled to a single-columnocean mixed layer model. A brief description ofthe key features of the model follows. A morecomplete description of the dynamical core equa-tions and discretization is given in Smith et al.(2015).

4.1.1 The atmosphere

The atmospheric component of the coupledmodel solves the primitive equations for prognos-tic variables, temperature, T , specific humidity,q, and ageostrophic zonal, u, and meridional, v,wind. The model is forced externally by horizon-tal advection for each of the prognostic variablesand by the geostrophic component of the winds.Tendencies due to sub-grid scale physical pro-cesses are also included to represent the effectof radiation, turbulent mixing, moist convectionand clouds.

The vertical discretization of the equations forthe atmosphere component uses the hybrid ver-tical co-ordinate scheme developed by Simmons

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and Burridge (1981) to describe the atmosphereon 60 model levels. This allows for greater res-olution in the planetary boundary layer (maxi-mum resolution is ≈15m), with decreasing res-olution towards the top of the model domain(minimum resolution is ≈4km) at 0.1 hPa.

4.1.2 The ocean

The ocean mixed layer model is based on theK-Profile Parametrization (KPP) vertical mix-ing scheme of Large et al. (1994). The codewas originally developed by the NCAS Centre forGlobal Atmospheric Modeling at the Universityof Reading (Woolnough et al. 2007) and incor-porated into the ECMWF SCM code by Takayaet al. (2010).

The prognostic variables in the ocean are themean values of temperature, θ, salinity, s, andzonal, uo and meridional, vo currents. The KPPmodel describes mixing in the boundary layernear to the surface and mixing in the interiorocean. This includes the effects of shear instabil-ity, internal wave breaking in the interior of theocean and double diffusion. The model is forcedby solar irradiance at the upper boundary andby externally specified geostrophic currents.

The ocean model uses a stretched vertical gridof Takaya et al. (2010) with 35 levels from thesurface to a depth of 250m. The resolution isincreased in the upper layers in order to simulatethe diurnal SST variability; the top model layeris chosen to be 1m thick and there are 19 levels inthe top 25m. Some examples of the model levelheights/pressures for the atmosphere and oceanare given in tables 1 and 2 respectively.

model level model full pressure level (hPa)1

17 18.81522 54.62425 95.98030 202.23033 288.09339 501.63749 861.49756 995.055

Table 1: Locations of the reduced observationsin the atmosphere used in experiments in section55.4.

model level depth (m)

1 1.0005 5.27710 11.40616 20.17320 28.10023 37.36625 46.98527 61.49829 83.81831 118.21433 170.77835 250.00

Table 2: Reduced observation locations in theocean used in experiments in section 55.4.

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4.1.3 Atmosphere-Ocean coupling

Coupling of the atmosphere and ocean compo-nents of the model occurs at every time step (15minutes) via the exchange of latent and sensibleheat fluxes and surface momentum flux from theatmosphere to the ocean. The updated oceanmodel sea-surface temperature is passed back tothe atmosphere where it is then used in the com-putation of the atmosphere lower boundary con-ditions for the next step. Fluxes are estimatedfrom bulk formulae, with the method of Louiset al. (1981) used to calculate the transfer coef-ficients.

4.2 Assimilation and forecast models

The coupled model used for the coupled DA ex-periments and to produce the coupled forecastsis similar to that used in Smith et al. (2015). Incomparison with the truth model this has miss-ing physics in the atmosphere, representing justadvection, vertical diffusion and turbulent mix-ing. It also has a positive bias in the large scaleforcing of the horizontal advection terms for theatmosphere. In the ocean, perturbed parametersfor the diffusion parameters are used (details ofwhich are given in table 3) and there is no non-local mixing. The parameters that are perturbedeach affect the mixing in the erroneous model.However their combined effect is minimal com-pared to the errors propagating down from theair-sea interface.

The uncoupled models used by the uncou-pled DA experiments are the same as those usedfor the coupled forecast with the exception thatthere is no exchange in information betweenthe two components. Instead the surface fluxesneeded to force the ocean component and theSSTs needed to force the atmosphere component

are prescribed externally. In this study we con-sider two options:

1. poor BCs: BCs given by ERA interim (Deeet al. 2011) and Mercator (Lellouche et al.2013) reanalyses products. These productsare inevitably inconsistent with the ideal-ized model used in these experiments. Inaddition to this the SSTs obtained fromthe Mercator product have no diurnal cycle,with only a daily averaged value provided.

2. good BCs: BCs given by output from thetruth model, Mt(xt

0). The output is pre-scribed at every 6 hours with linear interpo-lation to provide BCs at intermediate times.

4.3 Illustration of model error

The model error that results from this set up isillustrated for a case study relevant for July 2014for a point in the NW Pacific (188.75◦E, 25◦N).The initial conditions are obtained by runningthe truth model for 1 day initialized by datataken from ERA Interim Re-analysis for the at-mosphere and Mercator Ocean reanalysis for theocean valid at 00:00UTC on the 2nd July 2014.Obtaining the initial conditions in this way en-sures that they lie on the true model ‘attractor’.Forcing fields are also calculated from these re-analysis products, specified 6 hourly throughoutthe forecast period, with linear interpolation be-tween these times. The true evolution of the at-mospheric and oceanic temperature over an in-tegration time of 4 days is shown in figure 2.

In figures 3 to 6 the NL and TL model errorfor temperature in the atmosphere and ocean areshown, computed using equations (13) to (15).If we assume that the entire state is observedat every time step then the NL model error is

11

name description true value value used inassimilation andforecast

RRINFTY Critical Richardson number for shear instability 0.8 0.7RDIFMIW background/internal waves viscosity(m2/s) 1.5×10−4 1.0×10−5

RDIFSIW background/internal waves diffusivity(m2/s) 1.5×10−5 1.0×10−5

RDIFMMAX max viscosity due to shear instability (m2/s) 5×10−3 1.0×10−3

RDIFSMAX max diffusivity due to shear instability (m2/s) 5×10−3 1.0×10−3

Table 3: Parameters modified to create model error in the ocean component of the coupled model.

equivalent to εH and the TL error is equivalentto εTL. In this case the TL error has been com-puted for a perturbation equal to the truth mi-nus the background to be used in the assimila-tion experiments presented in section 5.

In figures 3 and 4 we see that in the atmo-sphere the NL and TL model error are fairlyinsensitive to the coupling strategy above theboundary layer (level 50 and above) and withinthe boundary layer only small differences can beseen. This suggests that for our model set upthe lower boundary is not a great source of er-ror. This could also be expected in general as theocean acts as a ‘slave’ to the atmosphere and sochanges to the ocean will not have an immediateeffect on the atmosphere at short time-scales. Ineach case we see a large warm bias forming inthe NL model between levels 40-50, which cor-responds to approximately 1.2-4.7km. There isalso a cool bias developing at level 30 which cor-responds to approximately 12km, roughly thetop of the troposphere. Compared to the NLmodel error the TL model error is small (ap-proximately 20% of its value) and should reducethroughout the minimization procedure.

Figures 5 and 6 show that the behavior of themodel error in the ocean is much more sensi-

tive to the upper boundary, suggesting that theatmosphere is a large source of error over thetwo day forecast window. For the coupled NLmodel (used in the strongly and weakly cou-pled scheme) we see that there is a cool biasat the surface, peaking at the diurnal maximum(roughly 24 hours and 48 hours into the fore-cast), overlying a warm bias. This suggests thatin the coupled model the heat originating fromthe atmosphere is being mixed down too quickly.This is seen to result in a large warm bias atlevel 20 (approximately 25m) after 1 day, corre-sponding to the thermocline being deepened tooquickly. This hypothesis is consistent with theerror in the lower winds which are also seen tohave a positive bias (not shown) and are there-fore causing too much momentum to be passedto the ocean and hence too much mixing inthe upper ocean. Experiments (not shown) inwhich the uncoupled ocean model is run withthe true heat fluxes but erroneous momentumfluxes (either computed from the coupled erro-neous model or prescribed externally from theERA interim product) show that the errors inthe momentum fluxes explain a large part of theerrors in the NL ocean model component seen infigure 5.

12

atmos temp (K)

time (days)

level

0 1 2 3

20

30

40

50

60200

220

240

260

280

ocean temp (K)

time (days)

level

0 1 2 3

5

10

15

20

25 295.5

296

296.5

297

297.5

Figure 2: The simulated true evolution oftemperature in the atmosphere (top) andocean (bottom) over an integration time 4days.

For the uncoupled model, figures 5 and 6 showthe error in the ocean to be very sensitive to theprescribed BCs. In particular we note that whenthe error in the BCs is negligible (last panel) theNL and TL model error is substantially reduced.When poor BCs are used, the error in the oceanictemperature may be larger than in the coupledmodel, but because the errors are not dynam-ically coupled to the atmosphere they are of acompletely different nature to the errors in theocean component of the coupled model. Most

notably there is an underestimation of the mix-ing of the heat into the ocean caused by an un-derestimation of the prescribed surface momen-tum flux (not shown), whereas in the coupledmodel there is an overestimation of the mixingas discussed above.

In contrast to the atmosphere, the TL errorin the ocean (figure 6) is of comparable mag-nitude to the error in the NL model (figure 5).However, both are substantially smaller than theerrors seen in the atmosphere.

In both the atmosphere and ocean it is inter-esting to note that the error in the approximateTL used by the weakly coupled formulation isonly slightly larger than using the exact TL inthe strongly coupled formulation. The main dif-ference in the atmosphere occurs in the lowest 10levels (corresponding to the BL) where the errorsare seen to be maintained longer into the simu-lation. In the ocean the greatest differences areat around level 20 (corresponding to the ther-mocline). Although these difference are small,we will see in section 55.4 that in some circum-stances they can lead to differences in the bal-ance between the ocean and atmosphere analy-ses.

4.4 Assimilation experiment design

Within this section the details of the assimilationexperiments setup are given. The aim of theexperiments is to study the effect of the modelerror on the assimilation of observations. Weconcentrate on the case study presented aboveso that the link between the resulting analysisand the realization of the model error is explicit.Experiments using data from a June 2013 casestudy for the same location gave qualitativelysimilar results so are not included but give usthe confidence that our results are robust.

13

The experimental design is essentially a biasedtwin experiment, in which the truth is known(see section 44.1) and observations are made di-rectly from this known truth. A biased model(see section 44.2) is then used to assimilate theseobservations.

In the following experiments the backgrounderror covariance matrix B is assumed to be di-agonal. This is a large simplification but allowsfor a clean comparison between the three dif-ferent coupling strategies introduced in section2. The variances of the background error areestimated from the variance in a time-series ofmodel output as described in Smith et al. (2015).The background is then computed by generatingwhite noise with the background error variances,adding it to the true profile at 24 hours priorto the initial time of interest and running thecoupled forecast model forward 24 hours. Thisensures that the background profile lies on thecoupled model attractor. As the errors in thebackground will have grown throughout the fore-cast the background error variances are then in-flated so that they are consistent with the errorsin the background profile. The background errorstandard deviations are shown in figure 7.

In the following experiments observations aremade of the truth at every model level at 3hourly intervals in the atmosphere and at 6hourly intervals in the ocean. The spatial den-sity of the observations has been chosen to ac-centuate the effect of the model error, which asseen in (10) and (11) is greatest when the obser-vations play a dominant role in calculating theanalysis, implying that the effect of model errorwill be greater with denser observations, espe-cially when coinciding with the region of largemodel error. The frequency of the observationshas been chosen to mimic the reduced availabil-ity of observations in the ocean compared to the

atmosphere.

The observations, simulated from the truthmodel, are consistent with a prescribed errorvariance which is assumed to be known exactlyin the assimilation. The observation error stan-dard deviations, the values of which are plottedin figure 7, are constant for each variable and in-dependent of height. We note that humidity isnot observed.

The inner loop is stopped when the relativechange in the gradient is less than 0.001. Thenumber of iterations needed depends on the as-similation scheme used and the window length.In general more iterations are necessary withthe strongly coupled scheme. To improve con-vergence a simple preconditioning of the controlvector is used. Instead of minimizing (5) with re-spect to δx0 it is instead minimized with respectto a transformed variable equal to B−1/2δx0.Such a transformation is commonly used in oper-ational data assimilation (see, for example, Ban-nister (2008) and references therein). In addi-tional tests (not shown) it has been found thatfor all DA strategies three outer loop iterationsare sufficient for convergence, however resultswill also be shown for the weakly coupled schemein which only one outer loop is performed.

5 Assimilation results

5.1 Sensitivity to window length

It can be expected that the assimilation re-sults will be sensitive to the assimilation win-dow length. If no model error is present andthe TL error remains small then increasing thewindow length can be expected to reduce theanalysis error as more observations become avail-able for assimilation. However, in the presence ofmodel error the analysis error will increase in ac-

14

cordance with (10) and (11) as the model errorgrows throughout the window and the greaternumber of observations accentuates its effect. Infigures 8 and 9 the absolute analysis errors (com-puted from the difference from the true state) atthe initial time for the different coupling strate-gies are given for assimilation window lengths of6 and 48 hours.

In figure 8 we see an increase in the analysiserror of atmospheric temperature for all couplingstrategies as the window length increases (notethe change in the x-axis scale). This is mostnoticeable at levels 45 and 25-30 where the anal-ysis becomes significantly poorer than the back-ground (gray line). These levels coincide withthe large biases seen in the assimilation modelwhich develop after approximately 12 hours (seefigure 3).

In figure 9 we see that for the analysis ofoceanic temperature the error does not increasein the same way as the window length is in-creased. Instead it appears in places that theanalysis error reduces, particularly between lev-els 6 to 10, with the uncoupled DA schemes (redlines) showing the greatest reduction in error.This is consistent with the longer window lengthallowing for the observations to provide more in-formation about the true ICs and the diurnalcycle of the evolution of the mixed layer. Thefact that the uncoupled analyses (especially withthe good BCs) outperforms the coupled analyseswas expected from figure 5. The opposite is truecloser to the surface at around level 2 despitethe improvement in temperature at level 1. Thiscould be indicative of the inability of the un-coupled methods to utilize information from theatmospheric observations to constrain the oceananalysis.

It is interesting to note the difference betweenthe analyses using the weakly coupled scheme

when 1 and 3 outer loops are performed (greendashed-dot and green dashed lines respectively).We see that for both window lengths an im-proved analysis is given when only one outerloop is performed around levels 6-9. The rea-son for this could be due to the fact that themodel error due to the coupling is only experi-enced in the weakly coupled scheme during theouter loop update, and so more outer loops im-plies that the coupled model has a greater influ-ence on the analysis (see Smith et al. (2015)).Therefore performing fewer outer loops reducesthe effect of the model error allowing for a moreaccurate analysis.

We also observe differences between the cou-pled and uncoupled analyses at level 18, justabove the thermocline, where the coupled analy-ses are more accurate than the uncoupled. Fromfigure 5 we see that the model error in this regionpropagates down from the surface. Therefore thelarger analysis error seen in the coupled analy-ses at the surface down to level 10, for a 48 hourwindow, may in fact be correcting for the errordeeper down and hence allowing for the modelprediction in this region to become more consis-tent with the observations and the informationin the observations to be interpreted correctly.No scheme is able to correct the large error atthe thermocline due to the relatively small back-ground error variances in this region (caused bythe fact that there was little variation in thisfeature in the forecast used to estimate the back-ground error variances, see section 44.4).

5.2 Forecast error when initializedfrom the analyses

The potential for the coupled DA scheme to pro-duce a poorer ocean analysis in the presence ofmodel error has been demonstrated. However,

15

often the aim of data assimilation is not to pro-duce the most accurate initial conditions but themost accurate forecast. In this section we lookat the error in the forecast produced using thecoupled erroneous model when initialized by thedifferent analyses computed with the 48hr assim-ilation window in the previous section. Figure10 shows the error in the forecast of atmospherictemperature. The top two panels show the errorin the forecast using the erroneous model wheninitialized with the true ICs (left) and the back-ground (right). The other panels show the errorin the forecast when initialized with the differentcoupling DA strategies (due to the similarity be-tween the weakly coupled results when 1 and 3outer loops are performed, only the results with3 outer loops are shown). It can be seen thatthe error in the analysis at the initial time (seenin figure 8 in which the error in the analyseswas larger than for the background between lev-els 25 and 30 and at level 45 in the case of a48 hour window length) has helped to restrictthe growth of the warm bias around level 45 andthe cool bias around level 30 in the forecast ini-tialized by the analyses at later times, comparedwith using the true ICs. This is because, in or-der to minimize the cost function an analysis wasfound which gave a good fit to the observationsthroughout the 48hr assimilation window andnot just at the beginning of the window. Thiseffect of model error in variational data assimi-lation has been noted previously in the work ofWergen (1992) and Griffith and Nichols (2000).

In figure 11 the forecast error for the atmo-spheric variables is summarized by the root meansquare error (RMSE) averaged over all atmo-spheric levels, plotted as a function of time. Thereduction in the forecast error for temperatureinitialized using the analyses (colored lines) com-pared to the true ICs (black line) is clear af-

ter 12 hours into the assimilation window andis maintained throughout the 48 hour assimila-tion widow and the following 2 day forecast. Theforecast error is also seen to be mostly reducedfor the wind fields, which displays an inertial os-cillation in magnitude. Note that humidity is un-observed which explains why the forecast erroris not reduced when initialized with the analysescompared to initialing with the truth.

In figure 12 the error in the forecast of oceanictemperature when initialized by the differentstates is shown. The reduction in the forecasterror when initialized using the different analy-ses compared with initialization from the truthis not as clear as for the atmosphere. However,the forecasts initialized using the coupled DAstrategies (middle row) do result in an improvedforecast in the region of the thermocline com-pared to the forecast initialized with the truestate, which is particularly noticeable beyond2 days. This is because, although using thecoupled non-linear model in the assimilation re-sulted in a larger analysis error in some regionsof the ocean, it is consistent with the coupledmodel used to produce the forecast. Therefore,as in figure 10, errors at the initial time havehelped to restrict the growth of the errors dueto the imperfect model. It is interesting to notethat this is also the case for the weakly coupledanalysis with only 1 outer loop which was seento give a more accurate analysis of the oceantemperature than when more outer loops wereperformed (see figure 9). We can therefore spec-ulate that in this case even with just one outerloop the weakly coupled scheme, by linearizingaround the coupled trajectory, has allowed for ananalysis consistent with the coupled model. Theimprovement seen in the error in the thermo-cline suggests that the assimilation has reducedthe amount of heat being mixed down from the

16

surface. From figure 13 we can see that this isnot due to a substantial improvement in the mo-mentum fluxes. In fact the fluxes initialized bythe strongly coupled analysis are worse in manyplaces, and may support the idea that surfaceheat fluxes are compensating for errors elsewherein the coupled model, as found in the studiesof de Szoeke and Xie (2008) and Zheng et al.(2011).

The results presented in figure 12 clearlydemonstrate the advantage of using the coupledscheme over the uncoupled scheme in reducingthe error in the ocean forecast. Despite themodel error it is important that the assimilationand the forecast models are consistent with oneanother. In the next section we show that this isparticularly true when forecasting using the cou-pled system, as it is essential not only that theanalysis allows for a good fit to the observationsbut also that the atmosphere and ocean analysesare in balance with one another.

5.3 Initialization shock

A benefit of coupled DA in the absence of modelerror is its ability to reduce the occurrence ofinitialization shock by updating the atmosphereand ocean as a coherent system and in turn findan analysis which lies on the forecast model at-tractor. In previous work (Smith et al. 2015)initialization shock was found to be evident inthe first few hours of forecast of sea surface tem-perature (SST). We now examine whether cou-pled assimilation can still reduce the shock whenmodel error is present and so the true attractorand model attractor differ.

In figure 14, 72-hour and 3-hour forecasts ofthe sea surface temperature (SST) are given us-ing the coupled model initialized using the differ-ent assimilation schemes with a 48hr assimilation

window. It is clear that although the uncoupledanalyses are closer to the true SST at the ini-tial time they quickly deviate from the truth andhave a much less realistic forecast during the firsthour than the coupled analyses. Beyond the firstfew hours the forecasts initialized using the un-coupled analyses continue to be poorer than theforecasts initialized using the coupled analyses.We can therefore conclude that, in this case, thepresence of model error at the atmosphere-oceaninterface does not adversely affect the ability ofthe coupled DA to produce a state consistentwith the coupled forecast model. The fact thatthe forecasts initialized with the coupled analy-ses still have a large cool bias in the SST, com-parative to the bias when initialized by the un-coupled analyses, is most likely related to thereduction in the warm bias seen in the thermo-cline see figure 12.

5.4 Effect of strength of coupling in‘weakly coupled’ DA

The strength of the coupling in weakly coupledDA is controlled by the number of outer loopsperformed and the resolution of the observations(Smith et al. 2015). In the previous experiment(using 3 outer loops and dense observations) theweakly coupled scheme is seen to perform in asimilar way to the strongly coupled scheme.

Within this section we wish to understand howthe model error affects the analysis when thestrength of coupling in weakly coupled DA is re-duced. Unfortunately the number of outer loopsand the resolution of the observations also have asignificant impact on the analysis in other waystoo, making it difficult to perform a clean ex-periment showing just the effect of the reducedcoupling. For example reducing the number ofouter loops reduces the ability to find the mini-

17

mum of the cost function and reducing the num-ber of observations means that the effect of theobservations and model error on the analysis isreduced (see (10) and (11)), so that there will beless deviation from the background no matterwhich assimilation scheme is used.

Given these caveats the assimilation experi-ments are now repeated using sparser observa-tion (both temporally and spatially). In the at-mosphere the frequency of the observations ismatched to the 6 hour frequency of the oceans(the frequency of the ocean observations remainsat 6 hourly) and in both systems the verticalresolution of the observations is reduced to thelevels given in tables 1 and 2. With this set uplittle difference was seen in the results betweenusing 1 or 3 outer loops in the weakly coupledformulation. This is because reducing the obser-vations means that the effect of the outer loopupdate of the innovations is reduced. Thereforeall the results that follow use 3 outer loops.

In figures 15 and 16 we see that the analysis er-ror is more comparable to the denser observationcase with a 48 hour window (see figures 8 and 9)if we increase the window length to 96 hours. Inparticular we see that we still get a large spike inthe analysis error at level 45 in the atmospherictemperature and we see a reduction in the anal-ysis error (compared to the background error)when the uncoupled methods are used aroundlevels 5-10 in the oceanic temperature. This isbecause we are still assimilating a similar numberof observations so that the effect of the observa-tions and model error is still significant allowingfor the analysis to divert from the background.

We expect this setup to reduce the strength ofthe coupling within the weakly coupled schemebecause although observations are available fora longer period of time the spatial and tempo-ral frequency of the observations is reduced and

so there is less information in observation spaceabout the coupling between the atmosphere andocean. In figure 17 this is illustrated by againlooking at the initialized forecast of SST. Com-pared to figure 14, we see that the initializa-tion shock has increased for both the stronglyand weakly coupled schemes, but the differenceis much greater for the weakly coupled scheme,with the effect of the initial imbalance seen be-yond the first hour of the forecast.

These experiments illustrate the potentialrisks of the weakly coupled scheme in the pres-ence of significant model error. As it is in theouter loop that the observations and model arecompared, the discrepancy due to the coupledmodel error will be similar for both the stronglyand weakly coupled formulations. However, be-cause the uncoupled TL models used in the in-ner loop are inconsistent with the coupled NLmodel used in the outer loop, the weakly cou-pled scheme is unable to find an analysis incre-ment which allows for an agreement between theobservations and the model as successfully as thestrongly coupled scheme if there is not enough in-formation about the coupling from the observa-tions. This means that not only can we expect apoorer analysis with the weakly coupled schemebut also a larger forecast error and a greater ini-tialization shock.

6 Conclusions and discussion

There is strong motivation for the developmentof coupled DA methods, namely their ability toproduce a more balanced coupled analysis stateand to make better use of near surface observa-tions. However, a limiting factor in the imple-mentation of coupled DA is the model error inthe atmosphere which restricts the length of the

18

assimilation window that can be used to some-thing shorter than in an uncoupled ocean onlyscheme. Within this study we have aimed to giveinsight into the effect of lengthening the windowwhen model error is present to see if the benefitsof coupled DA are still evident. A summary ofour key findings follows.

The effect of the model error in coupled DAdepends not only on the nature of the error inthe coupled model but also on the coupling strat-egy used within the DA scheme. It is possible forerrors in the coupled system to introduce an er-ror in the ocean component near to the surfacewhich has faster timescales than in the uncou-pled ocean model (as seen in figure 5 comparingthe coupled model error to the uncoupled modelerror with good BCs). This new source of errormeans that the accuracy of the ocean analysismay be degraded using a coupled scheme com-pared to using an uncoupled DA scheme which(in the absence of this fast error growth) is ableto utilize a longer window length . This wasshown to be evident for a case study in whichwe found the errors in the analysis of the oceanto be smaller in some regions when using an un-coupled scheme and a 48 hour window (figure9).

The clear problem with an uncoupled scheme,however, is that despite allowing a smaller errorin the analysis of the ocean, the atmospheric andoceanic analyses are inconsistent with the fore-cast model. This means that the error growthrate in the coupled model forecast may actuallybe larger when initialized using the uncoupledanalyses and an initialization shock may becomeapparent in the forecast of the SSTs (see figure14).

With dense observations and 3 outer loops, theweakly coupled scheme was seen to perform ina very similar manner to the strongly coupled

scheme, responding in a similar way as the as-similation window was increased (figure 9) andreducing the error in the forecast by a similardegree (figure 12). When the number of outerloops was reduced to one and observations weredense, it was seen that weakly coupled schemegave a better analysis than the strongly coupledscheme in some regions due to the reduction ofthe impact of the coupled error in the outer loopcalculation of the innovations, although the cou-pling was still strong enough to allow for theanalysis to be consistent with the coupled modeland so the forecast was better than that ini-tialised from the uncoupled analyses. For ex-ample, the error in the thermocline was reduced(figure 12) and initialisation shock was smaller(figure 14). However, if the density of the obser-vations is reduced then the strength of couplingis also seen to reduce, but instead of tending to-wards the uncoupled scheme (as is the case whenno model error is present (Smith et al. 2015))it can give a much poorer analysis and forecastthan both the uncoupled and strongly coupledschemes. This is because, although the infor-mation in the observations about the couplingis reduced, the inconsistency between the obser-vations and the NL coupled model seen in theouter loop remains. Therefore, unlike with thestrongly coupled scheme which used the coupledmodel within both the inner and outer loop, theweakly coupled scheme is unable to find an up-date to the background state which allows forthe initialized model to become more consistentwith the observations. Therefore the problemof model error is likely to be much more prob-lematic in the weakly coupled schemes, whichare currently being implemented at centers suchas the UK Met Office and ECMWF, than ina strongly coupled scheme. To address this is-sue, the implementation of coupled DA has been

19

forced to use a short assimilation window length(6 hours at the Met Office and 24 hours at theECMWF).

To conclude, the benefits of a coupled DAscheme are still evident even in the presence ofmodel error. However, if the aim is to find thebest analysis then it is important to choose anassimilation window length in which the modelerror remains negligible. In practice this meanschoosing a window length consistent with an at-mosphere only assimilation which will severelylimit the number of ocean observations avail-able for assimilation. If the purpose is to givean improved forecast then using a longer win-dow length may help to reduce the model er-ror growth (particularly in the observed fields)by finding the initial conditions which limit it.However, in order to use coupled 4D-Var to itsfull potential it is necessary to take into accountmodel error in the assimilation, allowing for thewindow length and number of ocean observa-tions available for assimilation to be increasedand theoretically a more accurate analysis to befound. Allowing for model error in variationaldata assimilation greatly increases the complex-ity of the data assimilation problem and is anactive area of research (e.g. Fisher et al. (2011),Moore et al. (2011)).

One method, known as weak constraint 4D-Var, aims to estimate the model error along withthe initial conditions (Griffith and Nichols 2000;Tremolet 2006). In theory this allows for themodel to be corrected using the observations.However, in practice it is very difficult to obtainaccurate results from weak constraint 4D-Var,as it is necessary to have a good understandingof the elusive model error statistics (see Todling(2014) and references therein). This is particu-larly challenging in coupled data assimilation asthe model error statistics do not only need to

be specified for the ocean and atmosphere butan understanding of the cross-correlations is alsoneeded for strongly coupled DA. One possibilityis to only estimate the error in the atmosphericcomponent assuming that the error in the oceanis comparatively negligible and has its origins inthe atmosphere for the timescale of the assimi-lation window.

An alternative method is to use parameter es-timation as well as initial state estimation, es-sentially tuning the model parameters to give abetter fit to the observations via the assimilation.Kondrashov et al. (2008) argue that systematicerrors in many tropical ocean-atmosphere gen-eral circulation models are caused by incorrectparameter values. Even if the model error is notentirely due to erroneous parameter values, pa-rameter estimation can be used to reduce modelerror if the model is sensitive to the parametersand the observations are sufficient (Navon 1997).This idea was used by Liu et al. (2014) with theFast Ocean Atmosphere Model (FOAM) and acoupled ensemble adjustment Kalman filter (An-derson 2001). They successfully estimated thesolar penetration depth (SPD) in twin experi-ments. SPD is thought to be a parameter thatmay have significant impact on the surface cli-mate (see Liu et al. (2014) for references). Theyalso tried to estimate two additional parametersrelated to the momentum and latent heat fluxes.This was less successful due to the nonlinear rela-tionships between the parameters and state vari-ables weakening the correlations between fore-cast error and parameter uncertainty. Estima-tion of the bulk adjustment factors was also per-formed by Mochizuki et al. (2009) using a 4D-Var technique. This was found to reduce modelbiases in climatological fields.

With both the weak constraint and parameterestimation methods it is unclear if the estimates

20

of respectively model error and the parametervalues should be used in the subsequent forecast.If they are not used then the models used in theassimilation and the forecast will be inconsistent,and so, in a similar way to the uncoupled DAscheme, the analyses will not lie on the forecastmodel attractor. Hence the forecast error growthmay be large even if the analysis error is small.

Acknowledgements: This work has beenfunded by a grant from the Natural Environ-mental Research Council (NERC), grant numberNE/J005835/1 and as part of NERC’s supportfor the National Centre for Earth Observation(NCEO). The authors would like to thank Pe-ter Jan van Leeuwen for valuable feedback onan earlier version of the manuscript and PollySmith who helped to develop the coupled singlecolumn model and the 4D-Var code.

A Derivation of analysis errorcovariance and bias in thepresence of model error

The analysis was given in (9) in terms of thebackground, xb, the observations over the as-similation window , y, the generalized non-linearobservation operator, H and the Kalman gainmatrix, K. In the presence of model error themapping H is erroneous as described by (8). TheKalman gain matrix is therefore no longer opti-mal and this has an impact on the analysis error.Let εa be the analysis error when there is no er-ror present in H (i.e. H = Ht) and εa be theanalysis error when there is error present.

εa = xa − xt0

= xb − xt0 + K(y − H(xb))

= xb − xt0

+K(y − Ht(xt0) + Ht(xt

0)

−H(xt0) + H(xt

0)− H(xb))

= εb + K(εo + εH −Hεb)

= εa + KεH

(16)

If we assume that εb and εo are unbiased thenεa is also unbiased and the expected value of εa

is

E[εa] = KE[εH] (17)

as stated in (11).

Similarly if we assume that εb and εo are un-

correlated with εH, we can compute the analysiserror covariance as

E[(εa − E(εa))(εa − E(εa))T]

= E[(εa + KεH −KE[εH])(εa + KεH −KE[εH])T]

= E[εa(εa)T] + KE[εH − E[εH])(εH − E[εH])T]KT

= Pa + KE[εH − E[εH])(εH − E[εH])T]KT

(18)where Pa is the analysis error covariance if nomodel error were present, as stated in (10).

References

Alves, O., Y. Yin, L. Shi, R. Wedd, D. Hudson,P. Okley, and H. Hendon, 2014: A coupledensemble ocean data assimilation system forseasonal prediction and its comparison withother state-of-the-art-systems. EGU GeneralAssembly Conference Abstracts, 16, 9487.

Anderson, J. L., 2001: An Ensemble AdjustmentKalman Filter for Data Assimilation. Mon.Wea. Rev., 129, 2884–2903.

21

Arribas, A., M. Glover, A. Maidens, M. Peter-son, M. Gordon, C. MacLachlan, and coau-thors, 2011: The GlosSea4 ensemble predic-tion system for seasonal forecasting. Mon.Wea. Rev., 139, 1891–1910.

Balmaseda, M., and D. Anderson, 2009: Impactof initialization strategies and observations onseasonal forecast skill. Geophys. Res. Lett., 36,L01 701.

Bannister, R. N., 2008: A review of forecast er-ror covariance statistics in atmospheric vari-ational data assimilation. ii: Modelling theforecast error covariance statistics. QuarterlyJournal of the Royal Meteorological Society,134 (637), 1971–1996.

Barnett, T. P., N. Graham, S. Pazan, W. White,M. Latif, and M. Fl ugel, 1993: ENSO andENSO-related Predictability. Part I: Pre-diction of Equatorial Pacific Sea SurfaceTemperature with a Hybrid Coupled Ocean-Atmosphere Model. J. Climate, 6, 1545–1566, doi:http://dx.doi.org/10.1175/1520-0442(1993)006¡1545:EAERPP¿2.0.CO;2.

Bormann, N., A. Hernandez-Carrascal,R. Borde, H.-J. Lutz, J. A. Otkin, andS. Wanzong, 2014: Atmospheric MotionVectors from Model Simulations: Part I:Methods and Characterizations as Single-Level Estimates of Wind. Journal of AppliedMeteorology and Climatology, 53, 47–64.

Courtier, P., J.-N. Thepaut, andA. Hollingsworth, 1994: A strategy foroperational implementation of 4D-Var, usingan incremental approach. Q. J. R. Met. Soc.,120, 1367–1387.

de Szoeke, S. P., and S. P. Xie, 2008: Thetropical Eastern Pacific seasonal cycle: assess-ment of errors and mechanisms in IPCC AR4coupled ocean atmosphere general circulationmodels. J. Clim., 21, 392–409.

Dee, D., 1995: Testing the perfect-model as-sumption in variational data assimilation.WMO Symposium on Assimilation of Ob-servations in Meteorology and Oceanography,Tokyo.

Dee, D. P., M. Balmaseda, G. Balsamo, R. Enge-len, A. J. Simmons, and J.-N. Thepaut, 2014:Toward a consistent reanalysis of the climatesystem. BAMS, 1235–1248.

Dee, D. P., S. M. Uppala, A. J. Simmons,P. Berrisford, P. Poli, and co authors, 2011:The ERA-interim reanalysis: configurationand performance of the data assimilation sys-tem. Q. J. Roy. Meteorol. Soc., 137, 553–597.

Fisher, M., Y. Tremolet, H. Auvinen, D. Tan,and P. Poli, 2011: Weak constraint and long-window 4D-Var. Technical Memorandum 655,European Centre for Medium-Range WeatherForecasts, Reading UK.

Frolov, S., C. H. Bishop, T. Holt, J. Cummings,and D. Kuhl, 2016: Facilitating strongly cou-pled ocean-atmosphere data assimilation withan interface solver. Mon. Wea. Rev., 144, 3–20.

Galanti, E., E. Tziperman, M. Harrison,A. Rosati, and Z. Sirkes, 2003: A studyof ENSO prediction using a hybrid coupledmodel and the adjoint method for data assim-ilation. Mon. Wea. Rev., 131, 2748–2764.

Gratton, S., A. S. Lawless, and N. K. Nichols,2007: Approximate Gauss-Newton methods

22

for nonlinear least squares problems. SIAMJournal on Optimization, 18 (1), 106–132.

Griffith, A., and N. Nichols, 2000: Adjoint meth-ods in data assimilation for estimating modelerror. Flow, Turbulence and Combustion, 65,469–488.

Han, G., X. Wu, S. Zhang, Z. Liu, and W. Li,2013: Error covariance estimation for coupleddata assimilation using a Lorenz atmosphereand simple pycnocline ocean model. J. Cli-mate, 26, 10 218–10 231.

Jin, E. K., and Coauthors, 2008: Cur-rent status of ENSO prediction skillin coupled ocean-atmosphere models.Climate Dynamics, 31 (6), 647–664,doi:10.1007/s00382-008-0397-3, URLhttp://dx.doi.org/10.1007/s00382-008-0397-3.

Kondrashov, D., C. Sun, and M. Ghil,2008: Data assimilation for a coupled ocean-atmosphere model: part II: Parameter estima-tion. Mon. Wea. Rev., 136, 5062–5076.

Laloyaux, P., M. Balmaseda, D. Dee, K. Mo-gensen, and P. Janssen, 2015: A coupled dataassimilation system for climate reanalysis. Q.J. Roy. Meteor. Soc., doi:10.1002/qj.2629,URL http://dx.doi.org/10.1002/qj.2629.

Large, W. G., J. C. McWilliams, and S. Doney,1994: Oceanic vertical mixing: a review and amodel with non-local boundary layer parame-terization. Rev. Geophys., 32, 363–403.

Lawless, A., S. Gratton, and N. Nichols,2005: An investigation of incremental 4D-Var using non-tangent linear models. Quar-terly Journal of the Royal Meteorological So-ciety, 131 (606), 459–476.

Lawless, A. S., 2013: Variational data assimi-lation for very large environmental problems.Large scale Inverse Problems: ComputationalMethods and Applications in the Earth Sci-ences, Radon series on Computational andApplied Mathematics, M. J. P. Cullen, M. A.Freitag, S. Kindermann, and R. Scheichl, Eds.,De Gruyter, 55–90.

Lea, D., I. Mirouze, M. Martin, R. King,A. Hines, and D. Walters, 2015: Assessinga new coupled data assimilation system basedon the Met Office coupled atmosphere, land,ocean, sea ice model . Mon. Wea. Rev., inpress, doi:10.1175/MWR-D-15-0174.1.

Lellouche, J.-M., O. Le Galloudec, M. Drevillon,C. Regnier, E. Greiner, and co authors, 2013:Evaluation of global monitoring and forecast-ing systems at Mercator Ocean. Ocean Sci., 9,57–81.

Liu, Y., Z. Liu, S. Zhang, R. Jacob, F. Lu,X. Rong, and S. wu, 2014: Ensemble-basedparameter estimation in a coupled general cir-culation model. J. Clim., 27, 7151–7162.

Louis, J. F., M. Tiedtke, and J. F. Geleyn, 1981:A short history of the PBL parameterizationat ECMWF. ECMWF Workshop on Bound-ary Layer Parameterization. Reading, 25-27November 1981.

Lu, J., and W. W. Hsieh, 1998: On determininginitial conditions and parameters in a simplecoupled atmosphere-ocean model by adjointdata assimilation. Tellus A, 50A, 534–544.

MacLachlan, C., A. Arribas, K. A. Peterson,A. Maidens, D. Fereday, A. A. Scaife, andcoauthors, 2015: Global Seasonal ForecastSystem version 5 (GloSea5): a high resolution

23

seasonal forecast system. Q. J. Roy. Meteor.Soc., 141, 1072–1084.

Madec, G., 2008: NEMO ocean engine. Note duPole de Modalisation, Institut Pierre-SimonLaplace (IPSL): Paris.

Magnusson, L., M. Balmaseda, S. Corti,F. Molteni, and T. Stockdale, 2013: Evalu-ation of forecast strategies for seasonal anddecadal forecasts in presence of systematicmodel errors. Clim. Dyn., 41, 2393–2409.

Mochizuki, T., S. Nozomi, T. Awaji, and T. Toy-oda, 2009: Seasonal climate modeling over theIndian Ocean by employing a 4D-VAR coupleddata assimilation approach. J. Geophys. Res.,114.

Molteni, F., T. Stockdale, M. Balmaseda,G. Balsamo, R. Buizza, L. Ferranti, and coau-thors, 2011: The new ECMWF seasonal fore-cast system (System 4). European Centre forMedium-range Weather Forecasting TechnicalMemorandum., 656.

Moore, A. M., H. G. Arango, G. Broquet, B. S.Powell, A. T. Weaver, and J. Zavala-Garay,2011: The regional ocean modeling system(roms) 4-dimensional variational data assim-ilation systems part i - system overview andformulation. Progress in Oceanography, 91,34–49.

Mullholland, D. P., P. Laloyaux, K. Haines, andM. Balmaseda, 2015: Origin and impact ofinitialisation shocks in coupled atmosphere-ocean forecasts. Mon. Wea. Rev., in press,doi:http://dx.doi.org/10.1175/MWR-D-15-0076.1.

Navon, I. M., 1997: Practical and theoreticalaspects of adjoint parameter estimation and

identifiability in meteorology and oceanogra-phy. Dyn. Atmos. Oceans., 27, 55–79.

Rawlins, F., S. P. Ballard, K. J. Bovis, A. M.Clayton, D. Li, G. W. Inverarity, A. C. Lorenc,and T. J. Payne, 2007: The Met Office globalfour-dimensional variational data assimilationscheme. Q. J. R. Met. Soc., 133, 347–362.

Saha, S., S. Moorthi, H.-L. Pan, X. Wu, J. Wang,S. Nadiga, and coauthors, 2010: The NCEPclimate forecast system reanalysis. BAMS, 91,1010–1057.

Saha, S., S. Nadiga, C. Thiaw, W. Wang,J. Wang, Q. Zhang, and coauthors, 2006: TheNCEP climate forecast system. J. Climate, 19,3483–3517.

Shelly, A., P. Xavier, D. Copsey, T. Johns, J. M.Rodriguez, S. Milton, and N. Klingaman,2014: Coupled versus uncoupled hindcastsimulations of the Madden-Julian Oscilla-tion in the Year of Tropical Convection.Geophysical Research Letters, 41 (15),5670–5677, doi:10.1002/2013GL059062, URLhttp://dx.doi.org/10.1002/2013GL059062.

Siddorn, J., C. Harris, H. Lewis, and W. Ten-nant, 2014: Coupling strategy for weather.Met Office MOSAC and SRG Meetings 2014,http://www.metoffice.gov.uk/learning/library/publications/science/met–office–scientists/mosac.

Simmons, A. J., and D. M. Burridge, 1981:An Energy and Angular-Momentum Conserv-ing Vertical Finite-Difference Scheme and Hy-brid Vertical Coordinates. Mon. Wea. Rev.,109 (4), 758–766.

Smith, P. J., A. M. Fowler, and A. S. Law-less, 2015: Exploring strategies for cou-

24

pled 4D-Var data assimilation using an ide-alised atmosphere-ocean model. Tellus A, doi:10.3402/tellusa.v67.27025.

Smith, S. M., R. Eade, and H. Pohlmann, 2013:A comparison of full-field and anomaly inial-ization for seasonal to decadal climate predic-tion. Clim. Dyn., 41, 3325–3338.

Sugiura, N., T. Awaji, S. Masuda, T. Mochizuki,T. Toyoda, T. Miyama, H. Igarashi, andY. Ishikawa, 2008: Development of a four-dimensional variational coupled data assimi-lation system for enhanced analysis and pre-diction of seasonal to interannual climate vari-ations. J. Geophys. Res.., 113, C10 017.

Takaya, Y., F. Vitart, G. Balsamo, M. Bal-maseda, M. Leutbecher, and F. Molteni, 2010:Implementation of an ocean mixed layer modelin IFS. Technical Memorandum 622, Euro-pean Centre for Medium-Range Weather Fore-casts, Reading UK.

Thepaut, J.-N., P. Courtier, G. Belaud, andG. Lemaıtre, 1996: Dynamical structure func-tions in a four dimensional variational assimi-lation: A case study. Q. J. R. Met. Soc., 122,535–561.

Todling, R., 2014: A lag-1 smoother approach tosystem-error estimation: sequential method.Q. J. R. Met. Soc., doi:10.1002/qj.2460.

Tremolet, Y., 2006: Accounting for an imper-fect model in 4d-var. Quarterly Journal ofthe Royal Meteorological Society, 132 (621),2483–2504.

van Leeuwen, P. J., 2014: Representation er-rors and retrievals in linear and nonlinear dataassimilation. Q. J. R. Met. Soc., 141, 1612–1623.

Vanniere, B., E. Guilyargi, T. Toniazzo,G. Madec, and S. Woolnough, 2014: A system-atic approach to identify the sources of trop-ical SST errors in coupled models using theadjustment of initialised experiments. Clim.Dyn., 43, 2261–2282.

Vitart, F., and Coauthors, 2008: The newVarEPS-monthly forecasting system: Afirst step towards seamless prediction.Q. J. Roy. Meteorol. Soc., 134 (636),1789–1799, doi:10.1002/qj.322, URLhttp://dx.doi.org/10.1002/qj.322.

Waller, J. A., S. Dance, A. S. Lawless, N. K.Nichols, and J. Eyre, 2013: Representativ-ity error for temperature and humidity usingthe Met Office high-resolution model. Q. J. R.Meteorol. Soc., 140, 1189 –1197.

Weaver, A. T., J. Vialard, and D. L. T. An-derson, 2003: Three- and four-dimensionalvariational assimilation with a general cir-culation model of the tropical pacific ocean.part I: formulation, internal diagnostics, andconsistency checks. Mon. Wea. Rev.,, 131,1360???1378.

Wergen, W., 1992: The effect of model errorsin variational data assimilation. Tellus, 44A,297–313.

Woolnough, S., F. Vitart, and M. A. Balmaseda,2007: The role of the ocean in the Madden-Julian oscillation: implications for MJO pre-diction. . Q. J. Roy. Meteorol. Soc., 133, 117–128.

Zhang, S., M. J. Harrison, A. Rosati, andA. Wittenberg, 2007: System design and eval-uation of coupled ensemble data assimilation

25

for global oceanic climate studies. Mon. Wea.Rev, 135, 3541–3564.

Zheng, Y., T. Shinoda, J. L. Lin, and G. N.Kiladis, 2011: Sea surface temperature biasesunder the stratus cloud deck in the SoutheastPacific Ocean in 19 IPCC AR4 coupled generalcirculation models. J. Clim., 24, 4139–4164.

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time (days)

level

coupled

0 0.5 1 1.5 2

20

30

40

50

60

time (days)

level

uncoupled (poor BCs)

0 0.5 1 1.5 2

20

30

40

50

60

time (days)

level

uncoupled (good BCs)

0 0.5 1 1.5 2

20

30

40

50

60 −5

0

5

Figure 3: Nonlinear model error εH for atmospheric temperature (K). The differentpanels show the errors present in the four different coupling strategies, from leftto right: strongly coupled DA, weakly coupled DA, uncoupled DA (poor BCs) anduncoupled DA (good BCs).

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time (days)

level

strongly coupled

0 0.5 1 1.5 2

20

30

40

50

60

time (days)

level

weakly coupled

0 0.5 1 1.5 2

20

30

40

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60

time (days)

level

uncoupled (poor BCs)

0 0.5 1 1.5 2

20

30

40

50

60

time (days)

level

uncoupled (good BCs)

0 0.5 1 1.5 2

20

30

40

50

60−1

−0.5

0

0.5

1

Figure 4: Tangent linear model error εTL for atmospheric temperature (K). Thedifferent panels are as in Figure 3.

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time (days)

level

coupled

0 0.5 1 1.5 2

5

10

15

20

25

time (days)

level

uncoupled (poor BCs)

0 0.5 1 1.5 2

5

10

15

20

25

time (days)

level

uncoupled (good BCs)

0 0.5 1 1.5 2

5

10

15

20

25

−0.5

0

0.5

Figure 5: Nonlinear model error εH for oceanic temperature (K). The different panelsare as in Figure 3.

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time (days)

level

strongly coupled

0 0.5 1 1.5 2

5

10

15

20

25

time (days)

level

weakly coupled

0 0.5 1 1.5 2

5

10

15

20

25

time (days)

level

uncoupled (poor BCs)

0 0.5 1 1.5 2

5

10

15

20

25

time (days)

level

uncoupled (good BCs)

0 0.5 1 1.5 2

5

10

15

20

25 −0.4

−0.2

0

0.2

0.4

Figure 6: Tangent linear model error εTL for oceanic temperature (K). The differentpanels are as in Figure 3.

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0 1 2

20

30

40

50

60

temp (K)

level

0 1 2

x 10−3

20

30

40

50

60

q (kg/kg)

level

0 5 10

20

30

40

50

60

u−wind (m/s)

level

0 5 10

20

30

40

50

60

v−wind (m/s)

level

0 0.5

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10

15

20

25

ocean temperature (K)

level

0 2 4

x 10−3

5

10

15

20

25

salinity (psu)

level

0 0.05 0.1

5

10

15

20

25

u current (m/s)

level

0 0.05 0.1

5

10

15

20

25

v current (m/s)

level

Figure 7: Profile of background error standard deviations for each of the prognosticvariables (solid line) and the observation error standard deviations (dashed line).Note that humidity is unobserved.

31

0 1 2 3

15

20

25

30

35

40

45

50

55

60

atmos t (K)

leve

l

window length 6hr

0 2 4 6 8

15

20

25

30

35

40

45

50

55

60

atmos t (K)

leve

l

window length 48hr

background

strongly coupled

uncoupled (poor BCs)

uncoupled (good BCs)

weakly coupled (3 outer loops)

weakly coupled (1 outer loops)

Figure 8: Absolute error in the analysis of atmospheric temperature at initial timeusing the four different coupling strategies. The window length increases from 6hr(left) to 48hrs (right).

32

0 0.1 0.2 0.3

2

4

6

8

10

12

14

16

18

20

ocean t (K)

leve

l

window length 6hr

0 0.1 0.2 0.3

2

4

6

8

10

12

14

16

18

20

ocean t (K)

leve

l

window length 48hr

background

strongly coupled

uncoupled (poor BCs)

uncoupled (good BCs)

weakly coupled (3 outer loops)

weakly coupled (1 outer loops)

Figure 9: As in figure 8 but for ocean temperature.

33

0 2 4

20

40

60

time (days)

level

true ICs

−5

0

5

0 2 4

20

40

60

time (days)

level

background ICs

−5

0

5

0 2 4

20

40

60

time (days)

level

ICs from strongly coupled DA

−5

0

5

0 2 4

20

40

60

time (days)

level

ICs from weakly coupled

−5

0

5

0 2 4

20

40

60

time (days)

level

ICs from uncoupled (poor BCs)

−5

0

5

0 2 4

20

40

60

time (days)

level

ICs from uncoupled (good BCs)

−5

0

5

Figure 10: Forecast error in forecasts of atmospheric temperature produced using theerroneous coupled model initialized using different initial conditions. The top twopanels show results when the model is initialized using the true state (left) and thebackground state (right). The other four panels show results when the assimilationmodel is initialized using the analysis produced by the different coupling strategiesusing a 48hr window and 3 hourly atmospheric observations and 6 hourly oceanicobservations. The blue vertical line indicates the end of the 2 day assimilationwindow.

34

0 1 2 3 40

0.1

0.2

0.3

0.4

time (days)

rmse

atmos temp (K)

0 1 2 3 40

0.5

1

1.5x 10

−4

time (days)

rmse

atmos humidity (kg/kg)

0 1 2 3 40

0.2

0.4

0.6

0.8

time (days)

rmse

atmos u (m/s)

0 1 2 3 40

0.2

0.4

0.6

0.8

time (days)

rmse

atmos v (m/s)

background ics

ICs from strongly coupled

ICs from uncoupled (poor BCs)

ICs from uncoupled (good BCs)

ICs from weakly coupled (3ol)

Figure 11: RMSE during the 48 hour assimilation window and subsequent 48 hourforecast window using the coupled model for the atmospheric variables. Coloredlines are the RMSE in the forecast when initialized with the different analyses (seefigure 8 for legend), the black line is the RMSE in the forecast when initialized withthe truth.

35

0 2 4

5

10

15

20

25

30

35

time (days)

level

true ICs

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)

level

background ICs

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)

level

ICs from strongly coupled DA

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)

level

ICs from weakly coupled (3 outer loops)

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)le

vel

ICs from weakly coupled (1 outer loop)

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)

level

ICs from uncoupled (poor BCs)

−0.5

0

0.5

0 2 4

5

10

15

20

25

30

35

time (days)

level

ICs from uncoupled (good BCs)

−0.5

0

0.5

Figure 12: As in figure 10 but for ocean temperature.

36

0 20 40 60 80−15

−10

−5

0

5

10

time (hours)

surf

ace s

ensib

le

heat flux (

W/m

2)

0 20 40 60 80−120

−100

−80

−60

−40

−20

time (hours)

surf

ace late

nt

heat flux (

W/m

2)

0 20 40 60 80−0.04

−0.03

−0.02

−0.01

0

0.01

time (hours)

surf

ace u

−str

ess (

N/m

2)

0 20 40 60 800

0.02

0.04

0.06

0.08

0.1

0.12

time (hours)

surf

ace v

−str

ess (

N/m

2)

Figure 13: Forecasts of the surface fluxes when initialized using the different analyses(colored lines as in figure 8). These can be compared to the truth (thick black line).

37

0 10 20 30 40 50 60 70297.3

297.4

297.5

297.6

297.7

297.8

297.9

298

298.1

298.2

time (hours)

SS

T (

K)

0 1 2

297.6

297.8

298

SS

T (

K)

Figure 14: Forecasts of SST initialized using the different coupling DA strategies(colored lines as in figure 8). These can be compared to the truth (thick black line).Inset shows just the first three hours to highlight any initialization shock.

38

0 1 2 3

15

20

25

30

35

40

45

50

55

60

atmos t (K)

leve

l

window length 48hr

0 2 4 6 8

15

20

25

30

35

40

45

50

55

60

atmos t (K)

leve

l

window length 96hr

background

strongly coupled

uncoupled (poor BCs)

uncoupled (good BCs)

weakly coupled

Figure 15: Analysis error for atmospheric temperature using reduced observationswith a 48 and 96 hour assimilation window.

39

0 0.1 0.2 0.3

2

4

6

8

10

12

14

16

18

20

ocean t (K)

leve

l

window length 48hr

0 0.1 0.2 0.3

2

4

6

8

10

12

14

16

18

20

ocean t (K)

leve

l

window length 96hr

background

strongly coupled

uncoupled (poor BCs)

uncoupled (good BCs)

weakly coupled

Figure 16: As in figure 15 but for ocean temperature.

40

0 10 20 30 40 50 60 70 80 90297.2

297.4

297.6

297.8

298

298.2

298.4

298.6

time (hours)

SS

T (

K)

0 1 2297.5

298

298.5

time (hours)

SS

T (

K)

Figure 17: As in figure 14 but for a reduced observation resolution and a 96 hourassimilation window.

41