Flow‐dependent wind extraction in strong‐constraint 4D‐Var

In the process of data assimilation for numerical weather prediction, biases in the model and observations can induce spurious analysis increments, which degrade the quality of the analyses. For this reason, the feedback of atmospheric composition (e.g., ozone and aerosols) observations on winds through dynamical adjustment is typically disabled in operational 4D‐Var assimilation. This study investigates whether an increasing number of tracer observations could be exploited to constrain winds better in 4D‐Var. For an idealized case study using analytical strong‐constraint 4D‐Var with the 1D advection model of Allen et al. and accurate background tracer field, it is shown that coupled wind–tracer assimilation always improves the tracer analysis. It also improves the wind analysis, but only if the magnitude of the error associated with unrepresented or misrepresented tracer physical forcings is smaller than the magnitude of the difference between the nature‐run advection and the background‐tracer advection. In other words, the model needs to be a good approximation of the truth for successful wind extraction. Based on this criterion, a method for flow‐dependent 4D‐Var wind extraction is developed, which selectively activates the coupling between winds and tracers locally and temporally by altering the tangent‐linear tracer equation and the adjoint wind equation. The new method is implemented in the intermediate‐complexity incremental 4D‐Var model Moist Atmosphere Dynamics Data Assimilation Model (MADDAM) of Zaplotnik et al. Numerical experiments with MADDAM show that the new approach diminishes the occurrence of spurious wind‐analysis increments and can improve the accuracy of wind analyses.


INTRODUCTION
Operational centres, such as the European Centre for Medium-Range Weather Forecasts (ECMWF), now routinely assimilate atmospheric composition measurements to provide the initial conditions (ICs) for forecasts of aerosols and trace gases, such as ozone, carbon monoxide, and so forth (e.g. Hollingsworth et al., 2008;Benedetti et al., 2009;Flemming et al., 2009;Morcrette et al., 2009;Dragani and Mcnally, 2013;Rémy et al., 2019). The composition measurements can be used not only to constrain the tracer ICs but also to perform an inverse estimate of dynamical variables, including the horizontal wind. Several studies have demonstrated the potential of tracers to constrain winds, using both 4D-variational (4D-Var) and Kalman-filter data assimilation techniques (e.g. Daley, 1995;Daley, 1996;Riishøjgaard, 1996;Peuch et al., 2000;Peubey and McNally, 2009;Semane et al., 2009;Allen et al., 2013;Allen et al., 2014). For instance, in regions with poor or no wind observations, internal model dynamics in 4D-Var adjusts the flow properties by minimizing the mismatch between observed tracers and their simulated equivalents. However, in the absence of wind observations, alterations done to the wind field by assimilation of tracers may not lead to better wind analyses (Zaplotnik et al., 2020). Extracting wind information from atmospheric composition data would be particularly useful in the Tropics and stratosphere, where wind observations are relatively scarce and wind analyses remain uncertain (Baker et al., 1995;Žagar et al., 2008;Baker et al., 2014), despite the recent successes of the radio occultation Constellation Observing System for Meteorology Ionosphere and Climate data (Ruston and Healy, 2021) and the Aeolus wind profiles (e.g. Rennie and Isaksen, 2021;Žagar et al., 2021). However, in the ECMWF 4D-Var system, the feedback of aerosols and trace gases on wind is turned off. This means that the prognostic equations for the aerosol and ozone mass mixing ratios are not used as strong constraints and the background (or latest guess) wind is used to advect the tracer increments. This is done to avoid potentially unrealistic wind increments due to observation biases and model errors/biases under certain atmospheric conditions or in certain layers Han and McNally, 2010). Therefore, improving both models and observations is necessary to enable full wind tracing.
The tracer forecast error is largely associated with the physical and chemical processes, which are unresolved or parameterized with a high level of uncertainty and simplified for computational purposes. For example, ozone photochemistry is simplified by relaxing the fields towards their climatological value (Cariolle and Déqué, 1986;Cariolle and Teyssèdre, 2007). Furthermore, in the ECMWF 4D-Var for aerosol assimilation, the tangent-linear model (TLM) of the total aerosol mass mixing ratio and its adjoint model (ADM) describe mass-conserving processes such as advection, convection, and diffusion , without dry or wet deposition, gravitational settling, washout, and other complex physical and chemical processes. This choice is justified by the high computational costs, underdetermined aerosol assimilation problem, and highly nonlinear physical and chemical processes which typically include switches and thresholds that are not directly differentiable. Under certain conditions, this introduces a systematic error of non-negligible magnitude.
In contrast to the use of aerosol and trace gas observations, the impact of humidity-sensitive radiance observations on dynamical states such as wind, temperature, and pressure is turned on and has a significant and growing impact (Andersson et al., 1994;Andersson et al., 2007;Geer et al., 2008;Peubey and McNally, 2009;Geer et al., 2017). Peubey and McNally (2009) separated the impact of 4D-Var humidity-wind tracing on forecasts from the impact of background-error correlations between humidity and temperature (e.g. Holm et al., 2002), which in turn affects wind, and the dynamical impact of improved humidity fields (via improved stability, condensation, latent heating, clouds, etc.) Their experiments confirmed that humidity-wind tracing is the most important mechanism by which humidity observations improve the wind fields. However, the impact of humidity observations is not spatially uniform. For example, the 4D-Var humidity-wind tracing appears to be less effective in the Tropics at around 400 hPa pressure level than in the midlatitudes, which leads to increased errors in the short-range forecasts of winds (Geer et al., 2014). While this impact may be an artefact of the own-analysis-based verification, the reasons for degraded skill may be physical as well: (1) the spatial variability of specific humidity in the Tropics is much lower than in the midlatitudes and the winds are slower on average, resulting in lower humidity advection (Zaplotnik et al., 2020), and (2) the magnitude of physical forcings, that is, the condensation rate and associated diabatic heating, are largest in the tropical midtroposphere. A high advection rate in comparison with the rate of physical forcing is essential for successful wind extraction, as we show in this article.
Detangling the extraction process of wind information from humidity and aerosol observations in full-scale 4D-Var systems such as the Integrated Forecasting System (IFS) is difficult. This is where intermediate-complexity models become of great help: for example, the Moist Atmosphere Dynamics Data Assimilation Model (MADDAM) of Zaplotnik et al. (2018). Within this framework, Zaplotnik et al. (2020) showed that aerosols are (at least) theoretically even better tracers than specific humidity and ozone, as their spatial variability in the troposphere is several times greater than that of the specific humidity and also greater than the spatial variability of stratospheric ozone. However, aerosols are more sparsely observed and are also a relatively new component of the global observing system and numerical weather prediction (NWP) systems Morcrette et al., 2009). Perfect-model 4D-Var experiments in Zaplotnik et al. (2020) revealed that it may be difficult to extract winds from aerosol observations in a humidity-saturated environment unless accurate, high-resolution observations of all thermodynamic variables are available to constrain the flow. In such conditions, highly nonlinear moisture processes and aerosol wet deposition could result in the assimilation running into a positive feedback loop, deteriorating the wind analysis in a similar way to that observed in operational systems. For instance, in a strong precipitation event, wet deposition is an order of magnitude larger than the typical magnitude of advection. If the wet deposition parameterization is omitted from the assimilation forward model and the aerosol observations suggest negative concentration increments, the internal 4D-Var dynamics compensates for a lack of aerosols by changing the (poorly constrained) winds and setting up a convergent flow (Zaplotnik et al., 2020). The present work presents ideas that may overcome such problems.
In this article, we ask which flow conditions are favourable for wind extraction and when it is necessary to either completely turn off or relax the wind-tracer coupling. Derivation of a suitable criterion is provided using the simple analytical 4D-Var assimilation model of Allen et al. (2013). Based on that, we develop a method for flow-dependent 4D-Var wind extraction, where the winds and tracers are locally and temporally (de)coupled in the TLM and its ADM. The new method merges the best of both worlds: it (1) uses the tracer data to constrain winds in certain areas and (2) ignores the tracer data where they may deteriorate the wind analysis.
This article is structured as follows: in Section 2, we explore the 4D-Var assimilation with the wind-tracer feedback switched on or off in a simple framework, and derive a criterion for controlling this feedback. In Section 3, we apply the flow-dependent wind extraction method in an observing-system simulation experiment (OSSE) in the 4D-Var MADDAM framework. Last, a case study is performed, where the flow-dependent wind extraction is compared with fully coupled and uncoupled wind-tracer assimilation. Discussion, conclusions, and outlook are given in Section 4.

COUPLED, UNCOUPLED, AND WEAKLY COUPLED WIND-TRACER ASSIMILATION: A MINIMAL EXAMPLE
In this section, we examine the impact of wind-tracer feedback on 4D-Var assimilation using a minimal example of a 1D linear tracer advection. Our aim is to determine analytically a criterion for flow-dependent wind extraction, such that the wind extraction is turned on or off based on the properties of the atmospheric flow. To do this, we will compare three options within the 4D-Var assimilation of tracer observations: • coupled case, with wind-tracer feedback switched on, • weakly coupled case, with a reduced coupling between the tracer and wind field, and • uncoupled case, in which the wind-tracer feedback is switched off.
We begin by formulating the 4D-Var assimilation problem. Following Allen et al. (2013) [1, their section 2.1], the 4D-Var cost function for a single tracer observation y with an error standard deviation o at time t 1 > t 0 is and an incremental formulation reads where . The nonlinear forward model is denoted  t 0 →t 1 , and M t 0 →t 1 is its TLM. The nonlinear observation operator is  and H is its linearized counterpart. In the wind and tracer background-error covariance matrix B, the errors for different variables (tracer and wind) and between distinct grid points are assumed uncorrelated so that B is diagonal. The observation departure from the background is = y(t 1 ) −  t 0 →t 1 (x b (t 0 )). The optimal increment (an analysis increment x a ) is achieved when the cost function is minimal, that is, when the cost-function gradient is zero: where H T and M T t 0 →t 1 are the adjoints of H and M t 0 →t 1 , respectively. The analysis increment is thus and the resulting analysis is

Perfect-model case
We first consider the perfect-model case of linear 1D advection of a passive tracer with constant, homogeneous wind u. This model represents the evolution of tracer concentration c on a periodic longitudinal ( ) domain at fixed latitude 0 , which is governed by the following equation: Equation (6) is discretized using a forward-time centred-space scheme on a periodic domain with three grid points. The model state vector is then , u] T . For simplicity, we choose Δt = Δ = 1. Only one integration step from time t 0 to time of observation t 1 is performed, thus The nature-run model  t t 0 →t 1 (note the superscript "t") represents the evolution of simulated truth from time t 0 to t 1 , and this will serve as a reference to simulate observations for our 4D-Var assimilation experiments. The nonlinear forward model in the assimilation ( t 0 →t 1 ) is assumed perfect, meaning that the governing equations and the processes described are exactly the same as in the nature run (Equation 7), therefore  t 0 →t 1 =  t t 0 →t 1 . The associated TLM of the nonlinear model, linearized around the background trajectory, then reads The system of Equation (8) describes the change in tracer concentration increment c with time due to the advection of the tracer increment by the background wind u b and the advection of the spatially varying background tracer field c b by the wind increment u. Note an additional parameter ( ), which controls the "level" of coupling of wind and tracer at position in the assimilation: • = 1 for coupled assimilation, • 0 < < 1 for weakly coupled assimilation, and • = 0 for uncoupled assimilation.
If = 0, only the background flow is used for tracer transport in Equation (8).
The discretized TLM (Equation 8) in matrix form is given by and its adjoint is where

Solution for a single perfect tracer observation
To evaluate the solution for a single tracer observation, we utilize the following idealized setup.
• The truth at time t 0 is defined as , 1] T and is propagated forward in time to construct a perfect (error-free) tracer observation at the time t 1 and longitude 2 : where  = H = [0, 1, 0, 0]. No noise is added to the tracer observation, but in the data assimilation we assume the observation-error standard deviation of .4] T , meaning we assume an initial error in the background wind b = u b − u t = 0.4, while the background tracer field is error-free at the initial time t 0 , that is, . At time t 1 , the background and truth tracer fields differ due to advection with an inaccurate initial background wind.
• The tracer background-error standard deviations are the same as the observation-error standard deviation, that is, The analyses for coupled and uncoupled assimilation for the above set of conditions are computed according to Equations (4) and (5), respectively, and are shown in Figure 1. The analysis accuracy is measured by the normalized analysis error (NAE), which is defined as the mean absolute difference between the analysis and the truth divided by the truth. We distinguish between wind NAE (denoted NAE u ), and mean tracer NAE over the three grid points (denoted NAE c ), A NAE of 0% indicates a perfect match of the analysis with the truth. In this perfect-model case with a perfect background tracer field, we find that the coupled wind-tracer assimilation outperforms the uncoupled assimilation, with NAE u of 17% compared with 40% and NAE c of 2.7% compared with 6.4%. We continue by examining how the combination of tracer and wind background-error standard deviations ( u and c , respectively) affects the results of 4D-Var assimilation. We compare coupled, weakly coupled (with coupling parameter = 0.5), and uncoupled assimilations, and analyze the normalized analysis errors for both wind (NAE u ) and tracer (NAE c ). Our results, shown in Figure 2, reveal that the wind analysis is better (lower NAE u ) for coupled or weakly coupled assimilations compared with an uncoupled assimilation for any combination of u and c . Only in the limiting case, where u → 0 (bottom of Figure 2d,e), does NAE u become the same for all three cases, as the background wind is highly accurate and cannot be changed by the assimilation of tracer observations. In practice, this has the same impact on wind extraction as switching off the tracer-wind feedback. The NAE u for the uncoupled assimilation is not shown, as the wind is not altered in this case and NAE u is 40% for any u and c . NAE c for a coupled or weakly coupled assimilation is always lower than for an uncoupled assimilation (Figure 2a-c). NAE c for a weakly coupled assimilation is between those for coupled and uncoupled assimilations. In the limit of u → 0, NAE c is the same in all three cases, as the wind is not allowed to change and the tracer field is advected only by the background wind. Similarly, all three cases have the same NAE c when c → 0, as the tracer field is not changed by the observation. If the forward model in 4D-Var is perfect and the background tracer gradients are accurately represented, our findings suggest that a coupled wind-tracer assimilation performs the best.

Imperfect-model case
The representation of physical processes affecting the tracers in the models is far from perfect. We discuss this case by introducing into our original model (Equation 6), used to produce the nature run, the time-dependent forcing F( , t) representing physical and chemical processes, such as wet deposition or dry deposition of aerosols. The tracer evolution is now described as and the truth x t evolves as where  t t 0 →t 1 is the same as before (Equation 7) and F t (t 0 ) represents the nature-run physical forcing at time t 0 . The systematic forecast error is introduced by assuming that the nonlinear forward model in 4D-Var simulates only advection without physical forcing, following Equations (6) and (7), and its TLM is described by Equation (8). Such a setup mimics the trace gas/aerosol assimilation setup at major operational centres, where only mass-conserving processes (e.g., advection, convection, diffusion) are simulated in the TLM/ADM for 4D-Var , whereas the physical forcings are omitted.

2.2.1
Solution for a single, perfect tracer observation The setup for the experiment is the same as for the perfect-model case (Section 2.1), with the addition of the physical , 0] T in the nature run. The total tracer change in the nature run at point 2 can be written as a sum of The perfect observation at time t 1 and collocated with grid point 2 at longitude 2 is now After the assimilation of a single, perfect tracer observation, the error of the resulting wind analysis (Figure 3) is greater for coupled (NAE u = 69%) than for uncoupled assimilation (NAE u = 40%). However, similarly to the perfect model case, NAE c is lower in the case of coupled assimilation (13%) than for uncoupled assimilation (30.5%). These results imply that it may be worth turning off wind-tracer feedback to avoid degradation of the wind field, where the physical forcing error is significant. Note that, in our example, introducing forecast error to grid points 1 or 3 by additionally setting F t ( 1 ) and F t ( 3 ) to nonzero values in the nature run would not affect the wind analysis, as the tracer observation at 2 is only affected by forcing F t ( 2 ) over a single time step (see Equation 17).
The same conclusion as for the case with the imperfect model would also apply in the case of observation error of ( 2 , t 1 ) = y( 2 , t 1 ) − c t ( 2 , t 1 ) = 1.5 and forcing F t ( 2 ) = 0, which would yield the same y(t 1 ) as in Equation (17). The observation error thus has a negative impact on the wind extraction similar to the forecast error, accumulated over the time of the integration. The properties of coupled wind-tracer 4D-Var assimilation under varying physical forcing F t at grid point 2 are explored further by measuring the wind NAE as a function of u and c . The magnitude of the nature-run forcing varies as F t = F( 2 ) = −1, −0.5, −0.25, 0.25, 0.5, and 1, while the nature-run advection A t and the background advection A b remain as before, that is, A t = −u t = −1 and Figure 4 illustrates several interesting features of the wind NAE, which can be generalized as follows.
1. If the magnitude of the nature-run physical forcing, omitted in the assimilation forward model, is lower than or equal to the absolute difference between the nature run and the background advection, then the wind analysis for a coupled assimilation is better than or equal to an uncoupled assimilation, for any combination of u , c , and o : As an example, consider Figure 4a,d with F t = 0.25 and −0.25, respectively, and |A t − A b | = 0.4. The analysis thus corrects the background wind towards the truth (Figure 5a). 2. If the magnitude of the physical forcing is higher than the absolute difference between the nature run and the background advection, and if the physical forcing is of an opposite sign to the advection difference, then the coupled wind analysis is worse than an uncoupled analysis for any combination of u , c , and o : then NAE wind (coupled) > NAE wind (uncoupled).
As an example, consider Figure 4e,f with F t < 0 In practice, the wind analysis is driven further away from the truth (Figure 5b) during the assimilation. 3. If the magnitude of the physical forcing is higher than the absolute difference between the nature run and the background advection and the physical forcing is of the same sign as the advection difference, then the combination of u , c , and o determines whether the coupled or uncoupled wind analysis is better (whether the analysis increment corrects the background towards truth or away from it), that is, Let us consider Figure 4b,c as an example. To conclude, in the case study presented, the 4D-Var wind extraction is beneficial when the magnitude of the missing physical processes in the assimilation forward model with respect to the nature run is less than the difference between the nature-run advection and the background advection. The same rules  apply also to any other combination of A t and A b for example, A t > 0 and A b < 0, A t < 0 and A b > 0, A t < 0 and A b < 0-in addition to the examples shown in Figure 5.
So far, the experiments of this section have kept the forcing F t constant. How does the wind-analysis error change when the forcing varies? Let us consider the same setup as in the experiment presented in Section 2.2 ( u = 0.2, c = 0.1, o = 0.1, u t = 1, and u b = 1.4) and Figure 3, but with varying F t . We observe that both wind and tracer analysis errors grow linearly with increasing F t (Figure 6). The linear dependence can be shown by including into Equation (4). The conclusion is that, in this simplified perfect model case with perfect tracer background, the tracer analysis is always better (lower NAE c ) for coupled tracer-wind assimilation, in the case of both perfect and imperfect models, regardless of the magnitude of the erroneous F t . In contrast, the wind analysis for coupled tracer-wind assimilation is better only if the model is a good approximation of the truth (under the conditions discussed above). In Figure 6a, the blue triangle indicates a range of F t values, where the coupled assimilation is better than the uncoupled one (−0.4 < F t < 1). The range is consistent with the criteria in Equations (18)-(20). In the case of weakly coupled wind-tracer assimilation with coupling factor = 0.2, this range of F t values expands, however the tracer analysis quality diminishes (the black dashed line in Figure 6b goes towards the grey dashed line of NAE c for the uncoupled assimilation).

FLOW-DEPENDENT WIND EXTRACTION
Can we use the results from Section 2.2 to perform the flow-dependent wind extraction? In Section 2.2, we assumed that the assimilation forward model only describes advection but none of the physical processes, thus the physical forcing in the background was F b = 0. However, in general, F b ≠ 0, and the criterion (18) for successful wind tracing transforms into The wind tracing is expected to improve winds when the difference between the nature run and the background physical forcing is smaller than or equal to the difference between the nature run and the background advection. The truth, F t and A t , can be expressed as As the truth is unknown, the criterion | F t | ≤ | A t | can be evaluated by approximating the truth by the latest guess fields (denoted by subscript "g"): | F g | ≤ | A g |. The two terms can be evaluated using an ensemble-based system such as the ensemble of data assimilations (EDA: (Isaksen et al., 2010;Bonavita et al., 2012), hybrid ensemble/4D-Var (Clayton et al., 2013;Bonavita et al., 2016), or any alternative method (e.g. Brecht and Bihlo, 2023) that provides an estimate of the flow-dependent uncertainty of the tracer advection and physical forcing. With the help of the ensemble, a criterion for tracing becomes: where ⟨| A g |⟩ is approximated by the mean absolute deviation from the ensemble mean, that is, ⟨| A g |⟩ ≈ ⟨|A g − ⟨A g ⟩|⟩, and analogously ⟨| F g |⟩ ≈ ⟨|F g − ⟨F g ⟩|⟩. This way, the size of the truth increments F t and A t is naturally related to the ensemble spread of A g and F g : a greater ensemble spread most likely results in a greater increment. Where and when the criterion in Equation (23) is met, the wind-tracer feedback could be turned on (coupled assimilation of winds and tracers), otherwise it is turned off (uncoupled assimilation of winds and tracers). The criterion deduced is used to perform flow-dependent wind extraction from simulated tracer (aerosol) observations using an intermediate-complexity data assimilation model.

MADDAM forecast and assimilation system
MADDAM is a 2D-horizontal spectral atmospheric model with a 4D-Var data assimilation system (Žagar et al., 2004a;  Gill (1980); Davey and Gill (1987), which describes the dynamical response to diabatic heating in the Tropics. The fuel for it is provided by moisture, which is advected with the flow, which depends again on the location and magnitude of heat sources. The diabatic heating is assumed to have approximately half-sinusoidal vertical structure with a maximum in the midtroposphere, while the horizontal winds attain a cosinusoidal vertical profile (opposite direction of winds in lower and upper troposphere) and the vertical distribution of humidity is exponential, as prescribed as in Davey and Gill (1987). The forecast model consists of five prognostic equations, which simulate the horizontal variations of the midtropospheric potential temperature perturbation ( ), lower level wind components v = (u, v), specific humidity q (units g kg −1 ), and aerosol mass mixing ratio (units g kg −1 ): Q LH and Q denote the latent heating and other diabatic forcings, respectively. Frictional processes for the mass and momentum are parameterized by terms representing the Newtonian cooling and the Rayleigh-wind friction with relaxation time scale = 1∕ of 2 months. The depth of the troposphere above the boundary layer is H ≈14 km, the background potential temperature is 0 = 333 K, and N denotes the buoyancy frequency. Other constants have their usual meaning: g is Earth's gravity and f = y is the equatorial -plane approximation of the Coriolis parameter. The parameters E and C in Equation (26) denote evaporation and condensation rates, respectively. Condensation occurs whenever moisture convergence or other processes (e.g., evaporation) produce excess humidity as defined by the local saturation specific humidity q s via the Clausius-Clapeyron equation.
Moisture surplus is immediately precipitated, and the specific humidity is set to its saturation value. The precipitation rate is P = kC, where k is some constant scaling parameter associated with the prescribed vertical profile of humidity. The released latent heat Q LH warms the atmosphere and increases the saturation value q s according to the Clausius-Clapeyron equation. The aerosol processes in Equation (27) involve advection, dry deposition (parameter K d ), and wet deposition (K w , scavenging rate is Λ = K w P), without any sources. The advection (A) and physical forcing (F) are denoted in Equation (27) to be in line with notation in Section 2. While MADDAM consists of the processes described in Equations (24)-(27), the following processes are switched off completely in the present study: evaporation E, nonlatent-heating diabatic term Q, and dry deposition K d . The remaining processes are simulated, including in the TLM and ADM.
The forecast model is coupled with an incremental 4D-Var data assimilation system formulated following Courtier et al. (1994). This 4D-Var formulation aims at finding the optimal analysis by performing a series of minimizations of a quadratic cost function for small deviations from the latest guess state. The latest guess is then corrected by the optimal increment, then a new nonlinear evolution is computed and the process is repeated until convergence is achieved. In our 4D-Var model, the state increment x = x − x g is transformed (using balance operator L) to a new control vector , such that x = L , in order to simplify the minimization problem. Similarly, the guess increment x g = x g − x b is transformed into a new vector g . The minimization problem can be written as consists of dynamical d , moisture RH n , and aerosol part c . The formulation of d and RH n was described in detail in Zaplotnik et al. (2018), while the assimilation model for c is described thoroughly in Zaplotnik et al. (2020).

Modifying tangent-linear and adjoint equations
How to include time-and space-dependent parameter (x, y, t) into the strong-constraint 4D-Var with a 12-hr assimilation window? Let us consider a 2D nonlinear advection of passive tracer c with wind v = (u, v), that is, without sources, sinks and other physical processes (as in (Zaplotnik et al., 2020). The governing equations are as follows: The associated TLM equations are whereas the ADM equations read as follows: Here, v and c represent basic state (background) variables, which evolve according to the nonlinear model (Equations 29a and 29b), v and c are small deviations from the basic state, for which the linearized model is valid (Equations 30a and 30b). v * and c * indicate adjoint variables, which define cost-function sensitivities at time t = 0. An outer product is denoted by ⊗. A tracer observation c obs enters the adjoint system (Equations 31a and 31b) at observation time t > t 0 via term J∕ c = (c − c obs )∕ 2 o , and affects the zonal and meridional wind cost-function sensitivities at initial time t 0 through the ADM integration. The backward transport of the tracer adjoint variables (term −∇ ⋅ ( c * v)) spreads the impact of an observation across the domain backward in time. The stronger the background wind, the more remote the response. The tracer observations also affect the wind (through terms − c * ∇c + c∇ c * in Equation (31a)).
To allow flow-dependent tracer feedback on the wind, the spatially and temporally dependent function (x, y, t) is introduced into the adjoint equation (Equation (31a) as follows: For the adjoint to remain valid, the tangent-linear equation for the tracer (Equation (30b) should also be modified as The numerical implementation in MADDAM is slightly different, since the model implements the adjoint of the numerical procedure instead of the analytical adjoint. Furthermore, the winds in MADDAM are affected by humidity and temperature observations as well.

Experiment setup
The impact of simulated tracer observations on the wind and tracer analyses is evaluated using the case of a strongly nonlinear flow near saturation, shown in Figure 7a,b at 6 and 12 hr of the simulation. Precipitation (in rainbow colours) occurs mostly in the central and southwestern part of the domain. This figure is a result of the nature run (NR) simulated by MADDAM at 0.25 • resolution. OSSEs with 12-hr 4D-Var are performed as fraternal-twin experiments with the forward nonlinear model, TLM and ADM at 1 • resolution. There are no other differences between the NR and 4D-Var model setups. The control background (BG) simulation at matching times, presented in Figure 7c,d, shows significant differences in precipitating regions. An ensemble of background simulations is also carried out, which allows us to evaluate the criterion in Equation (23) for selective flow-dependent wind tracing. It shows locally increased spread in the tracer forecast field associated with the precipitation and related wet deposition (figure not shown). The background ensemble is generated by perturbing the control background field using the randomization method (Andersson et al., 2000). The method applies random normal perturbations on the complex elements of the control vector, which uses the implemented background-error variance spectrum (Zaplotnik et al., 2018(Zaplotnik et al., , 2020. In this way, 50 ensemble members are generated. The background-error standard deviations b of different fields (zonal wind, meridional wind, temperature, humidity, aerosols) are as prescribed in Table 1. They are uniform and static and not flow-dependent. However, our aim is not to generate a perfectly representative ensemble but rather to briefly capture the uncertainties in the precipitation amplitude and location and the associated uncertainties in the aerosol wet deposition, to be able to run the demonstration experiments with flow-dependent tracing. Therefore, the amplitude of the perturbations is not b but rather b ∕5. For each perturbed background run, the wet deposition parameter has a different Gaussianly distributed random value K w =  (1, 0.5), in order to simulate the aerosol model error. The background ensemble is then used to evaluate the flow-dependent wind-tracer coupling parameter . The tracer observations are simulated from the nature run, with Gaussian noise  (0, 2 o ) added to the nature-run value ( o is the observation-error standard deviation). The relative observation error o ∕c of the aerosol mixing ratio is 20%, with a minimum error of 1 g kg −1 . The tracer observations are densely spaced both spatially and temporally, with observation locations 2 • apart (i.e. at every second grid point) in every direction and updated every hour, a total of 13 times within the 12-hr assimilation window. The same set of observations is used in all assimilation experiments. The rest of the model setup for the assimilation experiments follows Zaplotnik et al. (2020). Five different assimilation experiments were conducted: a fully coupled wind-tracer assimilation (coupling parameter = 1), an uncoupled wind-tracer assimilation ( = 0), and three experiments with flow-dependent wind-tracer coupling. There are several ways to model the coupling parameter based on the criterion in Equation F I G U R E 8 Flow-dependent wind-tracer coupling parameter for (a) case 1 with spatial-dependent, time-independent, and binary 1 (x, y), (b) case 2 with spatial-dependent, time-independent and continuous 2 (x, y), and (c,d) case 3 with spatial-dependent, time-dependent and binary 3 (x, y, t) at 6-hrs (c) and 12 hr (d). Black tones indicate strong wind-tracer coupling while white tones indicate weak coupling.
(23), which relates the ensemble spread of the physical forcing ⟨| F g |⟩ and the advection spread ⟨| A g |⟩. Options considered include the following.
1. Binary approach, where attains the same binary (0 or 1) value for the duration of the assimilation window: 2. Constant approach, where remains the same for the duration of the assimilation window, at a value ranging from 0 to 1 of form where H(x) denotes the Heaviside step function and N is the number of time steps. 3. Time-dependent , with the value of at time t defined by To estimate the flow-dependent , we must compute the physical-forcing spread ⟨| F g |⟩ and the advection spread ⟨| A g |⟩ for each model point and time step. Figure 8 shows the spatial distribution of for the different models described above. In general, grid points with < 1 coincide with areas of precipitation, where tracers are subject to wet deposition. For cases (1) and (2), remains constant throughout the assimilation window length (Figure 8a,b), while it changes between integration steps in case (3) (as shown in Figure 8c,d). By definition, < 1 for the same number of grid points in cases (1) and (2), but in case F I G U R E 9 Analysis at (a,c,e) 6 hr and (b,d,f) 12 hr for (a,b) uncoupled wind-tracer assimilation, (c,d) flow-dependent ( = 1 (x, y)) wind-tracer assimilation, and (e,f) fully coupled wind-tracer assimilation. Total aerosol mixing ratio is coloured grey while the rainbow colours denote 6-hr cumulative precipitation. [Colour figure can be viewed at wileyonlinelibrary.com] (1) is set to zero for every grid point, where < 1 in case (2).
Other possibilities for modeling the coupling parameter may include using spatial smoothing to minimize the spectral ripples in the TLM and ADM integration or to prevent possible excitation of gravity-wave-like features in the model in the case of sharp gradients in . Temporal smoothing or averaging of the fields over time intervals, such as a 3-hr interval, could also be applied.
However, exploring these options was beyond the scope of the present study.

Results
The resulting analyses from the 4D-Var assimilation of tracer observations are shown in Figure 9 for three different experiments: uncoupled wind-tracer assimilation for the spatial distribution of 1 ), and fully coupled wind-tracer assimilation ( = 1, Figure 9e,f). In the uncoupled wind-tracer assimilation, only the tracer analysis is corrected by the tracer observations, with no feedback of tracers on winds. As a result, the wind vector field is identical to the wind field in the background (Figure 7c,d).
In contrast, in the fully coupled wind-tracer assimilation, the tracers affect winds directly and also indirectly influence other dynamical fields in the process of 4D-Var internal adjustment. While the wind fields are generally improved in the analyses, a notable feature of our fully coupled assimilation case is the presence of spurious analysis winds in the region of heavy precipitation at 6 hr, located between the Equator and 7 • N and 285 • and 295 • longitude (Figure 9e), or at 12 hr at the same latitude and around 5 • N and 335 • longitude (Figure 9f). In this case, 4D-Var internal dynamics establishes locally highly convergent flow, which acts to compensate for tracer loss due to excessive wet deposition. Such spurious wind analyses have been observed in the operational 4D-Var (Dee, 2008) and have been explained in detail in Zaplotnik et al. (2020). They can occur whenever there is a large mismatch between the truth and the model evolution, and the dynamics are not constrained by the direct observations. The spurious wind analysis is partially alleviated by turning off the wind-tracer coupling in regions where the mismatch is more likely to occur, that is, where there is larger spread of the tracer physical forcing compared with the spread of the tracer advection (as described in criterion (23)). For example, the heavy precipitation and associated wet deposition of tracers in the narrow equatorial belt are associated with significant spread in physical forcing, so 1 (x, y) is mostly 0 in that region (Figure 8). By turning off the wind-tracer coupling in this region, the winds are not impacted by the tracers there and can only be affected by the advection of wind sensitivity in the backward integration of the adjoint model.
The positive impact of flow-dependent wind extraction in this case study is illustrated further by the evolution of the 12-hr analysis root-mean-square error (RMSE) scores ( Figure 10). The wind RMSE is evaluated as where N is the total number of grid points in the presented domain, v a i denotes the analysis wind vector at specific grid point (i, ), and v t i represents the corresponding wind vector in the nature run (truth). ‖⋅‖ 2 is the Euclidean norm. The wind RMSE for flow-dependent wind-tracer coupling is significantly reduced compared with the uncoupled assimilation and slightly reduced compared with the fully coupled assimilation, particularly at the beginning of the assimilation window. The tracer RMSE for flow-dependent cases is generally similar to that of the fully coupled assimilation, but 30%-40% smaller than for an uncoupled assimilation. The uncoupled assimilation provided the worst results in both the present case and the simple 1D case in Section 2. The RMSE reduction for flow-dependent cases compared with the fully coupled assimilation is even higher when the RMSE analysis is applied only to the precipitation areas (not shown). Similar results were obtained for other case studies.
Whereas both fully coupled and flow-dependent tracing both improve winds and tracers with respect to the uncoupled analysis, the tracing also introduces some imbalance to the system, which results in larger RMSE of the potential temperature field and greater values of kinetic energy power spectral density at small spatial scales (not shown).

CONCLUSIONS AND OUTLOOK
We compared the effects of coupled, uncoupled, and flow-dependent assimilation of winds and tracers in strong-constraint 4D-Var in the presence of a tracer forecast error. The main novelty of our study is a new method for flow-dependent 4D-Var wind extraction from tracers. The method takes the best of coupled and uncoupled wind-tracer assimilation: (1) it performs wind tracing in regions where advection dominates and the impact of tracing is expected to be positive, and (2) it avoids tracing in regions where the uncertainty of the physical forcing is large: for example, due to aerosol dry/wet deposition, condensation/evaporation, and so forth. By modifying the TLM and ADM equations, the method performs flow-dependent tracer feedback on the wind, that is, only at times and locations where the uncertainties due to physical forcing are inferior to the uncertainties associated with dynamics, that is, advection. The uncertainties can be estimated from the background ensemble: for example, the Ensemble Data Assimilation system of ECMWF (EDA: Isaksen et al., 2010;Bonavita et al., 2012). We deduced the criterion for flow-dependent wind extraction using a simple analytical 1D tracer advection model and the strong-constraint 4D-Var analysis of Allen et al. (2013). In a case study with perfect observation and accurate background tracer gradients, we demonstrated that wind tracing improves the wind analysis, but only if the magnitude of the error associated with unrepresented or misrepresented tracer physical forcings is smaller than the magnitude of the difference between the nature-run advection and the background tracer advection. Successful wind extraction thus requires the model to approximate the truth closely. Furthermore, we found that the tracer analysis is always superior for coupled tracer-wind assimilation, irrespective of the tracer forecast error related to physical forcing. This result was confirmed by a fraternal-twin OSSE experiment using the intermediate-complexity 4D-Var data assimilation system MADDAM (Zaplotnik et al., 2018). In this simplified framework where the only difference between the nature run and the background model is the horizontal model resolution, switching on the wind-tracer assimilation produced a positive impact on the tracer analysis.
We have also demonstrated that the flow-dependent wind-tracer assimilation can prevent the occurrence of spurious wind-analysis increments and improve upon the fully coupled 4D-Var wind-tracer analysis. Our results can also be generalized for other tracers, for example, ozone and trace gases, as their local concentration tendencies are (similarly to aerosols) nonuniformly affected by advection and physical forcings. For example, the rate of ozone generation and removal due to photochemistry, that is, the ozone physical forcing rate, can occasionally be larger than the advection rate in the summer hemisphere and (likely) also less certain. Based on the results presented in this article, we do not expect tracing to be successful in such cases.
The flow-dependent wind extraction should not be confused with blocklisting tracer observations, for example, humidity-or ozone-sensitive radiances (Forsythe et al., 2014), despite the fact that both methods may achieve a similar impact on wind analysis. However, similarly, blocklisting tracer observations also discards useful information for the tracer analysis.
Extensive experiments with realistic models are needed in the future, if the method is to be implemented in the operational NWP system. Intended future studies include wind-tracer coupling in weak-constraint 4D-Var, where the coupling can be controlled more naturally by the wind and tracer model error.
The tracer data carry a significant, but currently largely unexploited, potential to constrain the dynamics. More work should be done towards bridging that gap! AUTHOR CONTRIBUTIONS Žiga Zaplotnik: conceptualization; data curation; formal analysis; investigation; methodology; software; validation; visualization; writing -original draft; writing -review and editing. Nedjeljka Žagar formal analysis; investigation; methodology; software; writing -original draft; writing -review and editing, funding acquisition, project administration and resources. Noureddine Semane: formal analysis; investigation; methodology; software; writing -original draft; writing -review and editing.