a. Model and forcing

The ocean model used in this study is the rigid-lid version of the Océan Parallélisé (OPA) OGCM of the Laboratoire d’Océanographie Dynamique et de Climatologie (LODYC; Madec et al. 1998). The model solves the primitive equations for horizontal currents u_h = (u, υ)^T, potential temperature T, and salinity S.¹ The equations are formulated in orthogonal curvilinear z coordinates and discretized using finite differences on an Arakawa C-grid. The configuration of the model used here covers the tropical Pacific from 30°S–30°N to 120°E–70°W. The zonal resolution is 1° and the meridional resolution varies from 0.5° at the equator to 2° at artificial solid boundaries at 30°S and 30°N. The vertical resolution is divided into 25 levels, with grid depths ranging from 10 m in the upper 130 to 1000 m in the deep ocean. More details about this configuration of the model can be found in Vialard et al. (2001).

In the present study, the model is forced over the 6-yr period from 1993 to 1998 using wind stress derived from a combined weekly European Remote Sensing (ERS) satellite–Tropical Atmosphere–Ocean array (TAO) wind product (Vialard et al. 2002), daily fields of heat flux and evaporation from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996), and monthly precipitations from the Global Precipitation Climatology Project (Huffman et al. 1997). The initial conditions on 1 January 1993 were obtained from a spinup of the model as described in Vialard et al. (2002). They showed that a good simulation of the sea surface salinity (SSS) could be achieved with the above forcing combined with a flux correction strategy. Since we are also concerned with an accurate estimate of the salinity field in this study, we will follow the same approach. Each experiment in this paper is first run with a relaxation both to weekly sea surface temperature (SST) analyses from Reynolds and Smith (1994) and to monthly climatological SSS from Levitus et al. (1994). A relaxation coefficient of −40 W m⁻² K⁻¹ is used for SST, which corresponds to a restoring time scale of 1 month for a depth scale of 50 m. The relaxation coefficient used for SSS was chosen to give an equivalent restoring time scale to that for SST. At the end of each experiment, the 1993–98 monthly climatology of the SSS relaxation term was computed and added as a flux correction to the freshwater surface forcing. The experiments were then repeated without relaxation to SSS (but still retaining relaxation to SST). This strategy avoids large drifts of the SSS, while leaving it free to vary at interannual time scales.

It should be noted that this set of surface forcing is different from the one used in Weaver et al. (2003) and Vialard et al. (2003). For this reason, results presented in those two papers can only be compared qualitatively with those presented here. The impact of the different forcing fields is addressed in section 4b.

b. Assimilation and validation datasets

The observations used for assimilation consist of in situ temperature measurements from the Global Temperature and Salinity Pilot Project (GTSPP) of the National Oceanographic Data Center, and are the same data used by Weaver et al. (2003) and Vialard et al. (2003). This dataset includes measurements mainly from TAO moorings and Expendable Bathy-Thermographs (XBTs), plus a limited number of CTD casts and drifting buoys. A manual quality-control procedure was used to remove suspect data.

Salinity observations are not assimilated in this study but are used for validation. The Levitus et al. (1994) climatology (hereafter referred to simply as Levitus) is used as an observational reference. While this dataset does not allow for finescale validations of the analyzed variability, it will allow us to diagnose gross errors in the salinity analysis. Some additional observations are obtained from the Johnson et al. (2002) dataset compiled at the Pacific Marine Environment Laboratory. This dataset includes approximatively 3000 CTD profiles irregularly distributed over the 1993–98 period.

Current data from the TAO array will also be used for validation in this study. A description of these data and associated calibration procedures can be found in McPhaden et al. (1998). The comparison between the model results and the current data was performed by generating model averages using the same sampling as available for the current data.

c. Assimilation method

The assimilation method used in this study is a First Guess at Appropriate Time (FGAT) version of 3DVAR (Weaver et al. 2003). A description of the method is given below in order to highlight the specific extensions made to the method in the present study.

Let x = (u^T_h,T,S)^T denote the ocean model–state vector. In what follows, it is important to distinguish x, which contains the 3D prognostic model variables, from the analysis vector w, which contains only those components of x that are to be estimated from the observations using the assimilation method. [The notation used in this section closely follows the recommendations of Ide et al. (1997)].

In the 3DVAR of Weaver et al. (2003), the analysis was univariate, the only analysis variable being the 3D temperature field (w = T). The analysis was univariate since the formulation of the background-error covariance matrix [𝗕_(w)] was univariate and since only temperature data were assimilated. In this study, we continue to assimilate only temperature data but now develop a multivariate formulation of 𝗕_(w) to include constraints between temperature and salinity. This is sufficient to generate a salinity analysis in 3DVAR in the absence of direct salinity observations (see section 2d). The analysis vector in this study is thus w = (T, S)^T.

Let w^b = (T^b, S^b)^T denote the background estimate of w at analysis time. In 3DVAR, the objective is to determine the vector w that leads to the best simultaneous fit, in a statistically weighted least squares sense, to the observation vector y^o (here temperature) and to w^b. While the 3DVAR analysis is static, the observations that are assimilated are usually distributed over a given time window t₀ ≤ t_i ≤ t_n; i.e., y^o = [(y^o₀)^T, …, (y^o_i)^T, …, (y^o_n)^T]^T where y^o_i denotes the observation vector at time t_i. The fact that the measurement time of the observation may be different from the time of the background can be a source of bias in the 3DVAR analysis. The FGAT version of 3DVAR is an incremental algorithm designed to overcome this problem by allowing the observations to be compared with the background state at their appropriate time.

In 3DVAR FGAT, we seek to compute an increment δw = w − w^b that minimizes a quadratic cost function of the form of J = J_b + J_o, where

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𝗱 = (d^T₀, …, d^T_i, .…, d^T_n)^T, where d_i = y^o_i − H_i[w^b(t_i)] is the innovation vector; H_i is the (possibly nonlinear) observation operator at t_i; and w^b(t_i) is the background estimate of w(t_i) obtained by integrating the nonlinear model from t₀ to t_i with w^b(t₀) as an initial condition. The matrix 𝗛 = (H^T₀, …, H^T_i, …, H^T_n)^T is a concatenated linearized observation operator with H_i defined such that H_i[w(t_i)] ≈ H_i[w^b(t_i)] + H_i δw(t_i) where δw(t_i) = w(t_i) − w^b(t_i). In the present study, H_i is linear, and consists of bilinear interpolation on each model level and linear interpolation between model levels. For TAO data, which are available as daily mean values, H_i also includes a time averaging over the 16 time steps that make up a model day. Here 𝗥 is an estimate of the observation-error covariance matrix. The observation errors are assumed to be uncorrelated in space and time so that 𝗥 is diagonal; the error variances are set to (0.5°C)² for TAO data and (1.0°C)² for all other temperature data. The formulation of the background-error covariance matrix 𝗕_(w) requires careful consideration and will be discussed in detail in the next section.

Note that the choice of the analysis time within the assimilation window is somewhat arbitrary. Here, we define the analysis time to be at t₀ so that the background state w^b ≡ w^b(t₀) and the increment δw ≡ δw(t₀). As discussed in Weaver et al. (2003), with this definition, the FGAT version of 3DVAR can be interpreted as a limiting case of incremental 4D-Var in which the tangent-linear model that is used to transport δw(t₀) through the assimilation window in 4D-Var is replaced by a persistence model δw(t_i) = δw(t_i−1).

The cost function is minimized iteratively using a gradient descent algorithm. The limited-memory quasi-Newton algorithm M1QN3 (Gilbert and Lemaréchal 1989) is used here with an exact line search to improve its efficiency for quadratic minimization problems. The convergence properties of the minimization are further improved by employing a preconditioning transformation v = 𝗨^−Iδw in (1), where 𝗕_(w) = 𝗨𝗨^T and 𝗕⁻¹_(w) = (𝗨^−I)^T𝗨^−I, so that J_b = v^Tv2. The generalized inverse 𝗨^−I is needed since 𝗨 is rectangular in this study (see section 2d).

The 3DVAR problem is solved directly in the space of the control vector v, and then transformed back into physical space using the inverse transformation

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Denoting v^a as the minimizing solution, the analysis increment will be given by δw^a = U v^a, which in this study will contain increments for both temperature and salinity; δw^a = (δT^a, δS^a)^T. Following Bloom et al. (1996), the model-state vector is then updated gradually with the analysis increment via a constant 3D forcing term applied to the model equations between t₀ and t_n:

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where M (t_i, t_i−1) is the nonlinear ocean model operator acting on x = (u^T_h,T,S)^T between t_i−1 and t_i, and

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I denoting the identity operator and n is the number of time steps in the assimilation window.

d. Multivariate formulation of 𝗕_(w)

The spatial and multivariate structure of the analysis increments in a 3DVAR analysis is governed by the formulation of the background-error covariance matrix. To see this, it is sufficient to consider the exact minimizing solution of the 3DVAR problem (1) (Tarantola 1987):

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In the present study, where only temperature observations are assimilated, the increment δŵ highlighted by the underbrace in (5) will have nonzero values only for temperature δŵ = (δT̂,0)^T and only at those grid points that are directly influenced by the adjoint of the interpolation operator (𝗛^T). It is thus the application of 𝗕_(w) to δŵ in (5) that is primarily responsible for spatially spreading the observational information. Moreover, 𝗕_(w) is the only mechanism by which information can be transferred from the observed (temperature) to the unobserved (salinity) variable.

Our general approach for modeling multivariate constraints in 𝗕_(w) follows closely that of Derber and Bouttier (1999). The background-error covariance matrix is defined by

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where E[ ] denotes mathematical expectation, and primed variables denote the difference between the background field and the “true” field we wish to estimate [this difference is assumed to be unbiased in (6)]. The term B_TT represents the univariate covariances for T′. Following Weaver et al. (2003), we model these T′ − T ′ covariances directly as B_TT = ∑_TC_TT∑_T, where ∑_T is a diagonal matrix containing estimates of standard deviations, and C_TT = C^1/2_TT(C^1/2_TT)^Tis a symmetric matrix of horizontal and vertical correlations. These univariate correlations are modeled using a diffusion operator (Weaver and Courtier 2001) and are approximately Gaussian. The diffusion operator contains a number of tunable parameters that control the length scales and degree of anisotropy in the correlation functions. The horizontal length scales are assumed to be a function of latitude and to be symmetric about the equator. The zonal and meridional length scales are 8° and 2°, respectively, at the equator, and 4° poleward of 20°N or S, with a linear transition between these values within the equatorial strip. The vertical length scales are taken to be a function of depth, being twice the model’s local vertical grid spacing to provide adequate smoothing between model levels. This gives vertical scales of about 20 m down to a depth of 100 m and increased values below this depth. The background temperature–error standard deviations (∑_T) have been (crudely) approximated by climatology associated with the natural variability of the temperature field in an integration of the model without data assimilation. These values of the correlation parameters and standard deviations are identical to those used in Weaver et al. (2003) and Vialard et al. (2003).

Substituting (6) into (5), it is clear that when only temperature observations are assimilated, only the first column of 𝗕_(w) is used in the computation of the analysis increment. The analysis increment would be influenced by the second column if direct observations of salinity were assimilated or if the application of the observation operator resulted in a transfer of information between temperature and salinity, which would be the case in 4D-Var when implicit in 𝗛^T would be the integration of the adjoint of the tangent linear of the ocean model.

Suppose now that T ′ and S′ can be related through a linear T–S relation of the form S′ = K_STT′. A discussion of the validity of this assumption and of the precise form attributed to K_ST is deferred until the next section. This relation can then be employed directly in (6) to represent the univariate covariances for S′ (B_SS), and the cross covariances between T ′ and S′ (B_ST = B^T_TS) in terms of the univariate covariances for T ′ (β_TT):

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which can be factored as

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where K = (I   K^T_ST)^T. Derber and Bouttier (1999) refer to 𝗞 as a linear “balance” operator. For consistency, we will adopt the same terminology even though, in the current context, the term “conservation” operator may be more appropriate since the T–S relations used to define the operator K_ST will be derived from principles of water mass conservation rather than of dynamical balance. In this sense, K has a much wider meaning and should refer to any (linear) constraint between variables that can be incorporated in the covariance matrix.

From (8) it is clear that 𝗕_(w) has rank equal to that of the block matrix B_TT and thus is singular. The null space in 𝗕_(w) is associated with “unbalanced” salinity errors; that is, those errors not described by the balance operator K_ST. This explains why the generalized left inverse (𝗨^−I) has been used in the preconditioning transformation defined earlier. From (2), the inverse of the preconditioning transformation is given by δw =𝗨v, where 𝗨 = 𝗞B^1/2_TT and B_TT = B^1/2_TT(B^1/2_TT)^T. Note that while δw = (δT, δS)^T contains components for both temperature and salinity, v = υ_T contains a single component corresponding to a nondimensional (control) variable for temperature.

By ignoring unbalanced errors in salinity (i.e., by setting S′_U ≡ 0, where S′ = K_STT ′ + S′_U), we are in effect assuming 𝗞 to be exact, and treating it as a strong constraint in the analysis (Lorenc 2003). This is reflected by the single control variable υ_T in the minimization problem. It is possible to introduce K as a weak constraint [e.g., as in Derber and Bouttier (1999)] by prescribing a nonzero variance to the unbalanced errors and introducing a new control variable (υ_SU) for unbalanced salinity. In the current framework, however, unbalanced salinity variables can effectively be discarded since neither the observations (temperature) nor the 3DVAR observation operator (interpolation) provide information to correct them. The only information to correct salinity comes from the balance operator. The temperature and salinity analysis problems are thus effectively decoupled: a univariate 3DVAR analysis can be performed first to obtain δT^a, and then the balance operator can be applied directly to δT^a to obtain δS^a. Therefore, in the current study, our approach closely resembles the postanalysis correction technique of TH99, and both techniques could be expected to give similar results. Including the T–S relation within the covariance formulation, either as a strong or weak constraint, will have advantages in more general applications, such as combined temperature and salinity data assimilation, as discussed in section 6b.

a. Description of the T–S relation

In this section we wish to design an operator K_ST that can explain as much correlation (or balance) as possible between temperature and salinity. Furthermore, this operator must be linear so that it can be applied within a covariance matrix. Following TH99 and Troccoli et al. (2002), we consider a scheme that employs local T–S relationships from the model background state to constrain salinity as a function of temperature.

Suppose that the model background state possesses a well-defined T–S relationship so that, at any grid point, salinity can be expressed as a differentiable function of the background temperature, S^b = S(T^b). For a temperature perturbation T ′, we can then expand S (T^b + T ′) in a Taylor series about the background temperature state T^b:

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The salinity perturbation S′ resulting from a temperature perturbation T ′ can thus be defined, to first order, by S′ = ∂S/∂T|_T=T^bT′. Regions where this relationship is not valid (e.g., in the mixed layer where T and S are weakly correlated) will be considered shortly. To evaluate the derivative in (9), we assume that local temperature and salinity variations occur predominantly through vertical displacements of water mass (TH99; Troccoli et al. 2002). By the chain rule, we can then approximate the derivative in (9) as

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where we have neglected the horizontal components of the derivative. From (10) and (9), we can interpret the adjustment of salinity in the inverse of the preconditioning transformation as follows. On each iteration of the minimization process, a new value of the control variable υ_T is estimated and converted into a temperature perturbation T ′ = ∑_TC^1/2_TTυ_T. Using the local gradient of the background T profile, this temperature perturbation is then transformed linearly into an equivalent depth-dependent vertical displacement of the profile: (Δz)_T = ∂z/∂T|_T=T^bT ′. Then, using the local gradient of the background S profile and the vertical displacement vector (Δz)_T, a salinity perturbation S′ = ∂S/∂z|_S=S^b(Δz)_T is computed, and used along with T ′ in the evaluation of the cost function and its gradient for the next iteration of the minimization. The vertical derivatives are computed numerically using centered finite differences applied directly to the background T–S fields. This procedure gives an estimate of the vertical derivative at the midpoint between T (and S) points. The derivatives at successive midpoints are then averaged to give an estimate of the vertical derivative directly at T (and S) points.

Water mass T–S properties are largely preserved in regions where isentropic processes dominate (e.g., in the tropical thermocline) and hence we can expect a T–S constraint to be beneficial there. Where nonisentropic processes are important, for example, in the mixed layer, temperature and salinity variations are uncorrelated and we should avoid applying a T–S constraint. To account for this possibility, we introduce a multiplicative parameter α in the T–S relation such that

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where K_ST = α∂S/∂T|_T=T^b. For simplicity, α is set to either zero or one, depending on various conditions in the background state. To take into account the weak correlation between T and S in the mixed layer, we set α to zero in regions where the background vertical mixing coefficient A^b_υ is strong (A^b_υ > 5 × 10⁻⁴ m² s⁻¹). Temperature and salinity are also expected to be weakly correlated in regions where temperature is well mixed but salinity is strongly stratified (e.g., in the barrier layer; Lukas and Lindström 1991). Setting α to zero when (∂S/∂z|_S=S^b)/(∂T/∂z|_T=T^b) > 1 psu (°C)⁻¹ was found to be sufficient to eliminate spurious salinity increments arising under such conditions.² To avoid numerical problems in computing (10) in regions of weak temperature stratification, we also set α = 0 when ∂T/∂z|_T=T^b < 10⁻³ °C m⁻¹. Finally, a local two-gridpoint Shapiro filter was used to smooth K_ST in each model level.

It is important to note the strong dependence of K_ST on the background state, both through the parameter α and the derivative ∂S/∂T. This state dependency in K_ST implies that, in a cycling 3DVAR system, the T–S constraint will evolve from one cycle to the next and thus will implicitly take into account the impact of the data assimilation from previous cycles in deriving the background salinity error covariances for the current cycle.

b. Reconstructing a density profile from temperature observations: A twin experiment

In this section we perform an identical twin experiment to illustrate the impact of the T–S constraint in 3DVAR when only temperature “data” are assimilated. For simplicity, we consider the problem of reconstructing a density profile using data sampled from the corresponding temperature profile at different depths. The synthetic data were extracted from a true state taken to be the model restart on 1 January 1993 (see section 2a). The background state for the twin experiment was taken from a model restart obtained by spinning up the model using different surface forcing fields [the same ones used by Weaver et al. (2003)].

The temperature “observations” were extracted at 11 different depths (between 0 and 500 m) from a profile located at 160°E on the equator. No noise was added to the observations. Figure 1 shows the temperature and salinity increments produced by 3DVAR and the corresponding true increments defined as the difference between the background and true profiles. The temperature analysis increment is in quite good agreement with the true increment, although it underestimates the amplitude of the increment particularly near the maximum and base of the thermocline, as well as in the mixed layer (Fig. 1a). The salinity analysis increment computed using the T–S constraint gives a good estimate of the true salinity increment below the mixed layer, although it slightly underestimates its amplitude in the lowest part of the thermocline (Fig. 1b). This could be linked to deficiencies in the T–S scheme but also to deficiencies in the temperature analysis itself as noted above. Within the mixed layer (above 70 m), the analyzed salinity increment is 0 since the T–S constraint is not applied there. Figure 1b suggests that it may be desirable to extrapolate the balanced salinity increment at the base of the mixed layer directly to the surface (as illustrated by the dotted line in Fig. 1b) in order to avoid a discontinuity there. While this is an interesting possibility, we have not explored it in the experiments that are presented in the remainder of the paper.

Note that similar twin experiments were performed at different locations and times and the results were qualitatively similar to those illustrated in Fig. 1.

a. The assimilation experiments

Two sets of 3DVAR experiments are performed using in situ temperature data. One (TS) uses the T–S constraint in 𝗕_w to produce a multivariate analysis of temperature and salinity on each assimilation cycle, whereas the other experiment (NOTS) does not use this constraint and thus produces a univariate analysis of temperature on each cycle. The assimilation window is 10 days in both experiments. A third experiment (CTL) does not assimilate data and will be referred to as the control run. The experiments are run over a 6-yr period from 1 January 1993 to 31 December 1998 with the surface forcing fields described in section 2a.

b. Impact of the new forcing on the mean state

The reason for adopting the new forcing strategy compared to that used by Vialard et al. (2003) was to provide a better constraint on the upper-ocean salinity content. Figure 2 shows the average salinity in the upper 50 m of the TAO region (10°S–10°N, 160°E–70°W) for the univariate 3DVAR experiment (EX3D) presented in Vialard et al. (2003) and for NOTS, which is a similar experiment except that it uses the new surface forcing fields. Note that the two experiments also start from slightly different initial conditions as they have been produced using different “spinup” strategies. While the upper-ocean salinity does not seem to display a significant drift in NOTS, it decreases by 0.4 psu by the end of the 6-yr assimilation period in EX3D, suggesting that the new strategy for the freshwater forcing improves the quality of the upper-ocean salinity field.

Another important modification of the forcing is the use of the ERS–TAO wind stress product (see section 2a). The difference between the 1993–98 average zonal wind stress in ERS–TAO and its ERS counterpart (not shown) is negative over most of the TAO region, indicating that the easterlies are associated with stronger zonal wind stresses in the ERS–TAO product. This difference reaches 0.02 N m⁻² locally in the east Pacific, which represents between 10% and 30% of the mean wind stress. Figure 3 illustrates the impact of this stronger wind product on the zonal surface currents. As already discussed by Vialard et al. (2003), the univariate 3DVAR using the ERS forcing displays an eastward surface current bias at the equator in the central and eastern part of the basin (Fig. 3a). Bell et al. (2004) have suggested that this current bias could be linked to an imbalance that develops between the prescribed wind stress and the zonal pressure gradient when the latter is constrained by assimilated temperature observations. Indeed, if the prescribed easterlies are too weak and the zonal thermal gradient is constrained to be close to the observed gradient, the resulting pressure field will tend to accelerate the currents eastward. Since ERS–TAO wind stress has a systematically stronger zonal component than the ERS wind stress, the results from NOTS and EX3D would seem to confirm this explanation. In NOTS, the eastward surface current bias at the equator is reduced by 0.2 m s⁻¹ (Fig. 3b).

c. Impact of the T–S constraint on the temperature and salinity mean state

In this section the set of analyses (CTL, NOTS, and TS) is examined to assess the impact of the T–S constraint on the model temperature and salinity mean state. The 1993–98 average temperature from TS and NOTS are very similar (not shown); applying the T–S constraint does not significantly change the temperature analysis. In both experiments the main impact of the assimilation of temperature data is to tighten the thermocline by adding colder water below it and warmer water above it (Vialard et al. 2003).

Vertical equatorial sections of the 1993–98 average salinity in the upper 300 m of the ocean are shown in Fig. 4 for the Levitus climatology, the two assimilation experiments (TS, NOTS) and the control experiment (CTL). The Levitus climatology (Fig. 4a) is our observational reference. In general, the three experiments reproduce the main salinity structures of Levitus. However, in NOTS, the salinity maximum around 150 m in the west of the basin is reduced to 35.0 compared to 35.2 psu in Levitus, CTL, and TS. This bias is the result of the univariate assimilation of temperature since it does not appear in CTL, and is in agreement with Vialard et al. (2003) who showed a decrease in the upper-ocean salinity content in their univariate 3DVAR experiment (EX3D). In TS, where a T–S constraint is applied, this bias disappears and the maximum pattern is restored.

The mean 1996 salinity profiles between 0 and 2000 m for TS, NOTS, and CTL are shown in Fig. 5. CTL is in good agreement with Levitus especially below 500 m. In NOTS, however, the water is too fresh above 400 m and too salty below (this is also evident in Fig. 4). This is consistent with Troccoli et al. (2002) who showed that spurious mixing can freshen the upper-ocean and increase salinity in the deep ocean as a result of univariate temperature assimilation. In both TS and the experiment of Troccoli et al. (2002), the mean salinity profile is improved when the salinity constraint is applied. The T–S constraint prevents the artificial freshening and saltening above and below 400 m, respectively.

Figure 6 displays the evolution of the monthly salinity content in the upper 300 m in the TAO region. After 6 yr, the upper-ocean salinity content in NOTS has decreased by 0.2 psu, this decrease being strongest in the first 2 yr of the experiment. This salinity drift is a particular aspect of the univariate 3DVAR analysis since it does not appear in CTL and is largely corrected for in TS. Still TS displays some differences from the Levitus value (up to 0.1 psu). The salinity variability in TS, as well as in CTL, is difficult to validate in the absence of interannual observations of the salt content. However, it seems to be too large to be realistic, being equivalent to a change in precipitation of 1.1 mm day⁻¹ during 2 yr over the whole TAO region.

d. Comparison to independent salinity observations

The comparisons above are only qualitative since the Levitus and model climatologies are not representative of the same period. We now make a more quantitative validation of the model salinity fields by comparing them to the (nonassimilated) salinity observations of Johnson et al. (2002) (section 2b). The comparison was done by computing the difference between the model background salinity field and the data at their precise measurement time and location using the 3DVAR observation operator. These differences will be referred to as the background-minus-observation [BmO ≡ H_i(x^b_i) − y^o_i)].

Figures 7b,c present, respectively, the 1993–98 average and standard deviation of the BmO over the TAO region from CTL, NOTS, and TS. For reference, the average of the observed salinity profile in the upper 500 m of the water column is also shown (Fig. 7a). In the upper 100 m, there is no clear improvement in TS with respect to NOTS in either the mean or standard deviation. This could be expected since the T–S constraint is not applied in the mixed layer, and thus can not directly affect salinity there. In this region, CTL is in better agreement with salinity observations than either TS or NOTS. Between 100 and 250 m, there is much better agreement with the salinity observations in TS than in NOTS, as can be deduced from the smaller standard deviation and absolute average of the BmO. This confirms quantitatively the improvement of the subsurface salinity structure already illustrated qualitatively in section 4c. Below 250 m, TS displays the smallest standard deviation of the BmO, while CTL displays the smallest mean of the BmO.

Figure 8 shows the average and standard deviation of the BmO in the upper 300 m of the TAO region as a function of time. This figure shows that the average improvement due to the T–S constraint seen in Fig. 7 is also true at almost any time during the experiment. In both Figs. 7 and 8, the average of the BmO exhibits a clear positive and negative salinity bias in CTL and NOTS, respectively (Fig. 8a). At most times, TS displays a smaller average and standard deviation of the BmO than those in NOTS. The standard deviation of the BmO for CTL is slightly smaller than that of TS (as in the top half of Fig. 7c). This suggests that the variability of the salinity content in CTL is slightly better than in the assimilation experiments.

By comparing with independent salinity observations, we have thus shown that, when assimilating only temperature observations, there is a clear improvement in the mean and variability of the salinity field when the T–S constraint is applied, but no obvious improvement over the control experiment in which no data are assimilated.

e. Impact of the T–S constraint on the mean circulation

Figure 9 shows the 1993–98 zonal surface current for TS. It can be seen in Fig. 9 that the spurious equatorial eastward currents of NOTS (Fig. 3b) are eliminated in TS. Figure 10 shows vertical profiles of the zonal currents from CTL, NOTS, and TS compared to current data at three equatorial TAO moorings (at 165°E, 140°W, and 110°W). There is good agreement between TAO and CTL except near the maximum of the undercurrent, which is underestimated in CTL. At 140°W, the surface currents in TS are improved compared to those in NOTS (Fig. 10b). However, the intensity of the undercurrent maximum in the central and eastern Pacific is not as well captured in TS as in NOTS. Both TS and especially NOTS display eastward currents that are too large below the core of the undercurrent at 140° and 110°W, which are probably associated with large vertical velocities below the thermocline in both experiments (Figs. 11b,c).

Figure 12 allows us to diagnose the origin of the current differences between TS and NOTS. It displays the 170°–120°W zonally averaged difference between TS and NOTS for mean temperature, salinity, dynamic height, and the zonal velocity fields (Figs. 12a–d). Also displayed is the geostrophic component of the zonal current (Fig. 12e), which has been computed directly from the dynamic height difference in Fig. 12c using a combined f-plane and β-plane solution (Picaut and Tournier 1991). The differences in zonal currents between TS and NOTS are seen to be reasonably well approximated by geostrophy. The dynamic height (and thus current) differences are predominantly explained by the salinity differences in Fig. 12b. The eastward bias in NOTS can be related to the erosion of the subsurface salinity maximum close to the equator (between 2°S–5°N and 50–150 m in Fig. 12b) in NOTS, which causes a depression in dynamic height in the upper 100 m (Fig. 12c) and thus, because of the weak value of the Coriolis parameter there, a relatively strong eastward surface current anomaly of up to 20 cm s⁻¹. The better preservation of the salinity structure in TS prevents the development of this spurious current. Similar processes are involved to explain the difference between TS and NOTS below the core of the undercurrent, between 150 and 400 m near 2.5°S (in Fig. 10c). In this case, it is rather the subsurface salinity changes south of 5°S between 100 and 200 m in Fig. 12c that lead to the current differences.

a. Temperature budget

Figure 13 shows the 1993–98 average contribution to the total heat budget at the equator of zonal and vertical advection, vertical diffusion, and the temperature analysis increment for CTL, TS, and NOTS [see Eq. (A6) in the appendix]. Lateral diffusion and meridional advection do not contribute significantly to the average heat budget and are therefore not shown.

In CTL, the balance is representative of the thermocline equilibrium close to the equator. The zonal flow associated with the undercurrent carries warm water eastward, which results in a positive zonal advection term in the thermocline region (Fig. 13a) that is largely balanced by equatorial upwelling of deep cool water (Fig. 13b). These two contributions compensate to maintain the thermocline. In the eastern Pacific, where the thermocline is close to the surface, the influence of mixing becomes important (Fig. 13c). There, vertical diffusion tends to cool the surface and warm the subsurface.

This balance is modified in the presence of data assimilation since there is now an extra term in the heat budget coming from the heat added or removed by the analysis increment (see the appendix). In NOTS, the basic balance in the thermocline described above largely holds: warming by zonal advection, cooling by equatorial upwelling, and vertical redistribution of heat by mixing in the eastern part of the basin. However, most of the terms become notably stronger in the presence of data assimilation. Compared to CTL, the thermocline is tighter both in NOTS and TS, which results in larger horizontal and vertical temperature gradients. The undercurrent is strongly intensified in NOTS compared to CTL (Fig. 10). This results in a larger (and probably better) estimate of the zonal advection term that is approximately balanced by an increase of cooling by vertical advection (in turn linked to the vertical gradients and equatorial upwelling that are both stronger in NOTS and TS than in CTL as shown in Fig. 11). The increase in vertical mixing in the eastern Pacific is probably a result of both a tighter thermocline and spurious mixing in the assimilation experiments (Troccoli et al. 2002; Vialard et al. 2003). These contributions to the heat budget do not compensate as in CTL, thus resulting in local imbalances that are compensated by the analysis increment (Fig. 13g). The increment in NOTS has locally large values, with systematic warming or cooling of up to 3°C month⁻¹. The spatial structure of the increment can be recognized in parts of the vertical advection term (e.g., the strong cooling between 100 and 250 m around 175°W). At these locations, the assimilation leads to an unrealistic increase in vertical currents close to the equator (Fig. 11b). Figure 13 illustrates that this in turn leads to an increase in vertical advection that is perpetually counterbalanced by the analysis increment. As discussed in Vialard et al. (2003), the assimilation in a univariate 3DVAR continuously acts to correct for a bias of its own making. More generally, the assimilation compensates for any imbalance in the heat budget. For example, this is the case between 100 and 150 m at 140°W or at the surface in the eastern part of the basin where the origin of the imbalance is a combination of spurious zonal and vertical advection.

With the introduction of the T–S constraint the zonal advection term is reduced overall compared to that in NOTS (cf. Figs. 13d,h) and the vertical advection term is locally reduced where the salinity increment is large (cf. Figs. 13e,i). The thermocline structure is similar in both TS and NOTS suggesting that these reductions are the result of a weaker undercurrent (Fig. 10) and a weaker vertical current in TS (Fig. 11c). A positive feature of TS is that the mean temperature increment is reduced, indicating that the tendency of the assimilation system to produce a temperature bias through a spurious circulation has been reduced. The T–S constraint thus not only provides a better salinity state as illustrated in section 4c, but it also results in a better balance between terms in the temperature budget. It is also interesting to note that there is no significant change in the vertical mixing term between TS and NOTS, a point we will come back to in the next section.

b. Salinity budget

We now investigate the salinity budget to assess how the salinity increment is acting to prevent salinity biases from developing in TS. Figure 14 shows the main contributing terms to the salinity budget at the equator for CTL, NOTS, and TS. The largest terms are found close to the surface where the salinity gradients are strongest; in the temperature budget, they were found in the thermocline.

In CTL, the main equilibrium in the near-surface layer is between horizontal (zonal plus meridional) advection (Fig. 14a), vertical advection (Fig. 14b), and vertical mixing (Fig. 14c). Zonal advection carries freshwater westward close to the surface. This is in agreement with Vialard et al. (2002) who showed that this westward freshwater transport is due to both advection by the mean currents and transport by tropical instability waves. Equatorial upwelling (Fig. 11) combined with a negative salinity gradient between the surface and 100 m results in a salting above the subsurface salinity maximum (Fig. 14b). Finally, vertical mixing (Fig. 14c) tends to increase salinity in the fresh layer close to the surface, and freshen water in the region of the subsurface salinity maximum.

In NOTS, the biggest change in the zonal advection term (Fig. 14d) is in the subsurface central Pacific. It is probably the much stronger undercurrent in NOTS in this region that leads to a stronger estimate of zonal advection in this region. Salty water is advected from the maximum salinity region toward the east between 100 and 150 m. The upwelling below 100 m in NOTS (Fig. 11b), and thus below the subsurface salinity maximum, leads to a strong freshening between 100 and 200 m through vertical advection (Fig. 14e). The weakening of the subsurface salinity maximum that was seen in Fig. 4 also leads to a decrease in the upward advection of salt in the central Pacific between the surface and 100 m. At this depth, the vertical velocity is similar to that of CTL so that the change in vertical advection is due to the smaller magnitude of the vertical salinity gradient than that of CTL. Despite a noticeable increase in the magnitude of the vertical mixing coefficients in NOTS [not shown; see Fig. 8 in Vialard et al. (2003)], vertical diffusion is smaller in NOTS than in CTL because of the weakened salinity stratification in NOTS.

The terms of the salt budget are quite similar in TS and NOTS. However, the eastward advection of salt by the undercurrent and the subsurface freshening by spurious upwelling are stronger in TS. Both the undercurrent and the spurious upwelling are below 100 m are weaker in TS than in NOTS due to the stronger and better preserved salinity gradients in TS. It was seen earlier that the subsurface salinity maximum is better maintained in TS because there is a salinity increment that can act to compensate for the salt advection by unrealistic currents. Indeed, the salinity increment (Fig. 14j) has locally similar structures to those that appear in the zonal and vertical advection terms (Figs. 14g,h).

The description above is useful to understand how, in the stationary regime, the assimilation increment acts to compensate for imbalances between the physical terms of the salt budget. However it does not allow us to analyze the reasons for the salinity drift that was seen in Fig. 6. Figure 15 shows the time-integrated terms of the salinity equation over the upper 300 m of the TAO region [see Eq. (A8) in the appendix]. The vertical diffusion term has been split into the effect of surface forcing and the contribution from diffusive fluxes at 300 m [see Eq. (A9)]. In CTL (Fig. 15a) the change in salinity is mainly caused by surface forcing and advective effects. Horizontal diffusion contributes to a lesser extent, as a small but steady saltening term. These terms largely compensate each other, thus leading to a weak change of the upper-300-m salt content over the 1993–98 period (Fig. 6). In NOTS (Fig. 15b) the relative balance of terms in the salt budget is significantly different and results in the relatively large drift of −0.2 psu by the end of the period shown in Fig. 6 and by the dashed curve in Fig. 15b. This is essentially due to a change in the advective terms, which causes a strong freshening that is only partially compensated by the flux correction technique used in the freshwater surface forcing. This could be expected since the flux correction technique only has a significant impact near the surface. (Note that the use of flux correction implies that the freshwater forcing is different in CTL and NOTS.) Lateral diffusion does not change much between NOTS and CTL. On the other hand, the diffusive flux at 300 m, which was very weak in CTL, is now somewhat stronger and contributes to a 0.05-psu freshening over the whole period. Troccoli et al. (2002) invoked spurious mixing caused by static instabilities as the main cause of upper-ocean freshening in a univariate assimilation experiment similar to NOTS. The analysis above shows that the main cause of the freshening in NOTS is actually a change in advection, and that spurious mixing contributes to a much lesser extent. This difference may be partly linked to the absence of a vertical smoothing mechanism in the OI system of Troccoli et al. (2002), which could result in rather abrupt changes in the temperature field between levels and thus exacerbate the problem of artificial instabilities.

In TS the salinity drift over the whole period is significantly reduced (Fig. 15c). Compared to CTL, the only terms that change significantly are the surface forcing (due to the flux correction approach) and lateral diffusion, both of which contribute to a saltening that is approximately balanced by the salinity increment. While there was a strong change in advection in NOTS which led to a strong freshening of the upper ocean, the advective term in TS is qualitatively similar to that of CTL. This is in agreement with section 4e, which showed that the circulation was significantly improved in TS. The weak freshening of the upper layer by increased vertical mixing seen in NOTS is also noticeably reduced in TS.

a. Summary

Previous studies (Troccoli et al. 2002; Burgers et al. 2002; Vialard et al. 2003) have illustrated the importance of multivariate constraints to avoid generating spurious features in the tropical ocean salinity and current fields when assimilating subsurface temperature data. In this study, we have presented a general framework for introducing multivariate constraints in the background-error covariance matrix of variational data assimilation. Specifically, this framework was used to introduce a T–S constraint in the ocean 3DVAR system of Weaver et al. (2003). Following Troccoli and Haines (1999), the T–S constraint was defined by a local T–S relation derived from the model background state at the beginning of each assimilation cycle. A procedure was defined for linearizing the T–S relation so that it could be included as a constraint within the covariance matrix. By linearizing about the background state, the T–S constraint was made state dependent and thus could implicitly account for time variations in the local T–S relation.

An idealized single-profile experiment illustrated qualitatively how the method can reconstruct a salinity profile from temperature measurements. Three experiments were then performed over the 1993–98 period: a control run without data assimilation, and two 3DVAR experiments with and without the T–S constraint. In both 3DVAR experiments, in situ temperature data from the GTSPP were assimilated using a 10-day window. The incremental updating procedure of Bloom et al. (1996) was then used to correct the model fields with the analysis increment. With the T–S constraint, the 3DVAR analysis was multivariate in temperature and salinity. Without the T–S constraint, the 3DVAR analysis was strictly univariate in temperature.

Comparing the two 3DVAR experiments showed that the T–S constraint significantly improved the salinity field. This improvement in salinity complemented the improvement in temperature, which was similar in both assimilation experiments. The subsurface salinity maximum, which was strongly eroded in the univariate experiment, was better preserved when the T–S constraint was applied. Comparison to independent salinity observations confirmed that the salinity structure was improved, both on average and at most times during the experiments. However, the salinity variability in the multivariate experiment was slightly degraded relative to the control. The introduction of the T–S constraint also improved the ocean currents in the assimilation experiments. For example, it eliminated a large eastward bias in the surface currents in the central and eastern Pacific that was present in the univariate experiment. A similar current bias has also been observed in other univariate assimilation studies and has been loosely attributed to a general “balance” problem arising from univariate assimilation near the equator (Burgers et al. 2002; Vialard et al. 2003; Bell et al. 2003). In this study it was shown that this bias was associated with eastward geostrophic currents that were generated because of an erosion of the salinity maximum. In the multivariate experiment, the T–S constraint was able to restore the salinity maximum and thus reduce these artificial geostrophic currents.

Analysis of the heat and salt budgets in the various experiments allowed us to examine how the assimilation changed the term balances in the model. In the univariate experiment, the assimilation created a spurious circulation that perpetually degraded the model thermal structure. On each assimilation cycle, the analysis increment was then forced to counteract this problem by correcting the temperature bias that was inflicted by the assimilation in the previous cycles. When the T–S constraint was applied, the spurious circulation was reduced and the mean temperature increment decreased, suggesting that less work had to be done by the assimilation to correct for the bias of its own making. While the spurious currents were reduced in the multivariate experiment, they were still present and tended to degrade the salinity field. In the multivariate experiment, the salinity increment could compensate for this bias and thereby maintain the salinity structure, in a similar way to how the temperature increment could compensate for a related bias in the temperature field.

The analysis of the mean upper-ocean salt budget as a function of time allowed us to identify unrealistically strong advection in the univariate experiment as the main cause of the salinity drift, which led to a strong freshening of the upper ocean. In the multivariate experiment, the advection term was restored to a level comparable to that of the control run, and as a result the salinity drift was significantly reduced. It is interesting to contrast this explanation to that of Troccoli et al. (2002) who suggested enhanced vertical mixing resulting from univariate temperature assimilation as the main reason for the erosion of the salinity maximum (and hence freshening of the upper ocean). Vertical mixing was shown to be strong in the univariate experiment but not the dominant factor. Indeed, the improvement seen in the surface currents when the T–S constraint was applied cannot be explained by a reduction of vertical mixing since this would not affect the overall heat and salt content of the water column and thus would lead to only a small change in sea level and surface currents.

b. Discussion

One of the main objectives of this study was to illustrate how the nonlinear salinity adjustment method of Troccoli and Haines (1999) could be cast within the framework of variational data assimilation. A distinguishing feature of our approach was the use of a linearized T–S constraint within a balance operator of the background-error covariance matrix. Since the balance operator acts on increments generated during the minimization process, the linear assumption would not appear to be a particularly restrictive one, providing the increments are “small.” Since the improvements to our analyses were qualitatively similar to those documented in Troccoli et al. (2002), we suspect that the errors induced by linearization were minor in this study. The linear assumption may break down, however, if the increments are large in which case a relinearization of the balance operator about a recent estimate of T and S obtained during minimization may be desirable to improve the approximation.

While many aspects of the ocean analysis have been significantly improved with the inclusion of the T–S constraint, various problems remain. For example, the near-surface salinity is degraded relative to both the control and univariate experiments (Fig. 7), zonal and vertical currents below the thermocline are far too strong (Figs. 10 and 11), and vertical mixing coefficients are abnormally large (not shown). These findings suggest that further improvements to the assimilation system are necessary.

The increased errors in the near-surface salinity may be partly related to the discontinuity in the T–S constraint at the base of the mixed layer, where it is set to zero. Rather than applying no constraint on salinity between the base of the mixed layer and the surface, it may be beneficial to extrapolate the salinity balance just below the mixed layer directly up to the surface. This would be a straightforward extension of the method that should be tested. The analysis of the salinity field could obviously be improved through the direct assimilation of salinity data from, for example, Argo floats and CTDs. The general strategy adopted in this study for modeling T–S constraints in the covariance operator makes it straightforward to assimilate salinity data in combination with temperature data, and thus to create a two-way flow of information between temperature and salinity. Better use of salinity data could be made by using weak constraint versions of the T–S constraint so that balanced covariances can be given less weight relative to “unbalanced” covariances in regions where T–S relationships are weak. These extensions would require careful tuning of the covariance statistics of the unbalanced component of the background salinity error to achieve an appropriate partition of the covariances into balanced and unbalanced components.

Our multivariate formulation of the background-error covariance matrix gives a critical role to the covariance statistics of (balanced) temperature for establishing the balanced part of the covariance matrix for salinity. Improvements in the specification of the temperature error covariances would thus translate into improvements in the analysis of salinity as well as temperature. There is considerable scope for improving these covariance estimates. For example, the climatological variances that were used for the background errors are probably too large and too broadly spread out over the upper ocean. This may induce increments that are too large below the thermocline and be partly responsible for the overly strong vertical mixing and vertical velocities seen there. A better representation of the variances would take into account the local stratification of the thermocline so that the larger variances can be properly focused in regions where the local variability is strongest. One way to achieve this is to parameterize the variances in terms of the vertical gradient of the background temperature state. Improvements to the velocity field can also be expected by extending the balance operator to include geostrophic constraints (Burgers et al. 2002), thereby creating a fully multivariate 3DVAR analysis that can simultaneously correct all state variables (temperature, salinity, and velocity) from any single data type. Recent 3DVAR sensitivity experiments with these new developments have indeed resulted in a noticeable decrease in the magnitude of the vertical velocities and improvement in the zonal currents. These developments to the covariance matrix can also be expected to have a positive impact in 4D-Var.