Do Nudging Tendencies Depend on the Nudging Timescale Chosen in Atmospheric Models?

Nudging is a ubiquitous capability of numerical weather and climate models that is widely used in a variety of applications (e.g., crude data assimilation, “intelligent” interpolation between analysis times, constraining flow in tracer advection/diffusion simulations). Here, the focus is on the momentum nudging tendencies themselves, rather than the atmospheric state that results from application of the method. The initial intent was to interpret these tendencies as a quantitative estimate of model error (net parameterization error in particular). However, it was found that nudging tendencies depend strongly on the nudging time scale chosen, which is the primary result presented here. Reducing the nudging time scale reduces the difference between the model state and the target state, but much less so than the reduction in the nudging time scale, resulting in increased nudging tendencies. The dynamical core, in particular, appears to increasingly oppose nudging tendencies as the nudging time scale is reduced. A heuristic analysis suggests such a result should be expected as long as the state the model is trying to achieve differs from the target state, regardless of the type of target state (e.g., a reanalysis, another model). These results suggest nudging tendencies cannot be quantitatively interpreted as model error. Still, two experiments aimed at seeing how nudging can identify a withheld parameterization suggest nudging tendencies do contain some information on model errors and/or missing physical processes and still might be useful in model development and tuning, even if only qualitatively.

time scale, τ ndg . This approach is referred to as observational nudging. Another possibility is that the target state is gridded. In this case, the nudging term often takes the following form: where u 0 = u 0 (x, y, z, t) is some gridded target state (e.g., analysis, model output) spatiotemporally interpolated onto the host model grid and W = W(x, y, z, t) is a weighting function that can be used to limit nudging to certain regions, model levels, or times. This approach is often referred to analysis nudging, though, any gridded state can be used as a target. Here, analysis nudging is the focus with the weighting function set to unity (W = 1).
The use of such nudging first appears in scientific literature in the mid-1970s (Anthes, 1974;Davies & Turner, 1977;Kistler, 1974). The original application was to use nudging to assimilate data for analyses and model initial conditions. This approach more continuously assimilated information from data, preventing numerical instabilities that arose in previous assimilation methods that simply inserted observations (e.g., Charney et al. (1969); Jastrow and Halem (1970)). For additional early historical context and applications, see Stauffer and Seaman (1990) and Skamarock et al. (2008).
Subsequently, another application of analysis nudging was first demonstrated in Kao and Yamada (1988), where a numerical weather model was nudged to analyze observations, allowing realistic tracer transport and diffusion with synoptic-scale meteorology constrained. Such an exercise was sometimes preferable to just using the standard 12-hourly analyses of the time, allowing physical processes represented in the model to influence the state trajectory and tracer advection/diffusion between analyses in a plausible way. Currently, using analysis nudging to constrain meteorology in advection, diffusion, and chemistry models is fairly common, often referred to as a "specified dynamics" configuration of some host model (e.g., Davis et al. (2021)). A more nuanced and recent application of the method is to use nudging to constrain part of the atmosphere (e.g., the stratosphere) and seeing how the rest of the atmosphere (e.g., the troposphere) responds (e.g., Simpson et al. (2011); Hitchcock and Simpson (2014); Tripathi et al. (2015)). This list of nudging applications is likely not comprehensive.
Here, the initial intent was to use nudging to approximate errors in model parameterizations, with particular interest in errors in the momentum equations. As nudging keeps the host model's state close to a presumably accurate target state, one might be tempted to interpret the nudging tendencies time averaged over, say, a season, to be the net error in total (i.e., dynamical core + all parameterization tendencies) model forcing per unit mass (Equation 1). If one were to further assume dynamical core (i.e., discretization) errors are significantly smaller than those due to inaccurate/missing representations of under-and un-resolved processes (i.e., physical parameterizations), then the nudging tendencies might be interpreted more specifically as the net error of all parameterizations. Then, nudging tendencies and physical understanding of relevant processes at play could be used to attribute errors to particular processes, allowing these errors to be quantitatively targeted. Mapes and Bacmeister (2012) use similar arguments to interpret incremental analysis update tendencies as net error of all physics parameterizations. Analysis increments, used to correct a model with a data assimilation system toward collections of observations, have also been used to similarly comment on model errors (e.g., McLandress et al. (2012)).
However, while evaluating such an approach, it was found that nudging tendencies are proportional to the inverse of the nudging time scale (   −1   ) chosen and, therefore, cannot be interpreted as total model (dynamics + physics) error or net physics error, at least not quantitatively. The sole objective of this article is to present this apparently unpublished, if not completely unknown, result. The ubiquity of nudging capabilities and their numerous uses within weather and climate models motivate presentation of this result.
Here, nudging tendencies required to keep the Community Atmosphere Model, version 6 (CAM6, Gettelman et al. (2019)), close to the ERA5 reanalysis (Hersbach et al., 2020) over five recent winters are analyzed, with nudging time scales varied from 3 to 24 hr. The model configuration and experiments are described in Section 2. The primary result that nudging tendencies are proportional to −1 is presented in Section 3. In Section 4, a heuristic theoretical argument for this result is provided. Two experiments testing the extent to which nudging can highlight withheld parameterization tendencies are presented in Section 5. A summary and discussion of the results is given in Section 6.

Model Configuration and Experiment Design
The default CAM6 was used to simulate five boreal winters beginning 1 Dec from 2008 to 2012 with specified sea surface temperatures and sea ice (i.e., the "FHIST" component set was specified) on a ≈ 1° latitude/longitude grid with the finite-volume dynamical core. The model was initialized every 1 Dec with ERA5 interpolated onto its grid and integrated for three months (i.e., over December, January, and February (DJF)). Regridding was performed using the Betacast software package (Zarzycki, 2022;Zarzycki & Jablonowski, 2015). During the integrations, CAM was continuously nudged to ERA5 winds, temperatures, and humidities at every time step. With a time step of 30 min, the target state, ϕ 0 = (u 0 , v 0 , T 0 , q 0 ), was updated to ERA5 at the top of every hour and to ERA5 linearly interpolated in time half-past every hour. Four such simulations of these five winters were completed, each with a constant nudging time scale of τ ndg = 3, 6, 12, and 24 hr.
All tendencies acting on the zonal and meridional winds were output. Exact budget closure was verified. Here, generally, the net tendencies by the dynamical core and by all physical parameterizations are presented, often referred to as "dynamics" and "physics" tendencies. Figure 1 shows the DJF-averaged net dynamics, physics, and nudging tendencies projected along the DJF-averaged wind horizontal wind vector vertically averaged over the lowest four levels. For example, the nudging tendencies in Figure 1d were computed via Here, a nudging time scale of τ ndg = 6 hr was used.
where 〈.〉 is a DJF, 2008-2012 time average and ⟨ ⟩ is a unit vector pointing in the direction of the DJF, 2008-2012 time averaged horizontal wind vectors.

Typical DJF Momentum Balances
Here, the focus is on the lowest four model levels, as an initial interest was low-level wind biases that nudging tendencies at these levels highlight. Generally, the dynamical core acts to accelerate the low-level flow (Figure 1a), while the physical parameterizations (the turbulence and low-level drag parameterizations in particular) oppose these dynamics tendencies and act to decelerate the low-level flow (Figure 1b). The dynamics and physics tendencies do not exactly cancel, however ( Figure 1c). The residual of these two tendencies is almost exactly compensated by the nudging tendencies. This is expected, as ∂ t u ≈ 0 after time averaging over DJF, requiring the sum of the dynamics/physics residual tendencies and the nudging tendencies to be small as well (Equation 1). The total wind time-averaged tendencies are an order of magnitude smaller than those in the bottom row of Figure 1 (not shown).

Do Nudging Tendencies Change With τ n ?
To get a sense for how robust nudging-tendency-derived model error estimates were, nudging time scales were varied while keeping the model configuration (default except for nudging) unchanged. The momentum nudging tendencies, , are shown in Figure 2 for four runs with τ ndg varied from 3 to 24 hr.
Two results are immediately apparent: the spatial structure of the momentum nudging tendencies are quite consistent across the four runs and the nudging tendencies increase with decreasing τ ndg . These results held when looking at the middle troposphere and lower stratosphere as well (not shown). Nudging tendencies tend to exert additional drag over regions with significant topography (e.g., the Andes, the Rocky Mountains, the Himalayas, the flanks of Antarctica) in all four simulations. These tendencies also exert drag on the low-level winds in all four runs over the Southern Ocean, with tendencies somewhat invariant in longitude at these latitudes. Regions where the model might be exerting too much drag, with nudging tendencies accelerating the low-level flow, also display some consistency across the runs, though, perhaps less so than the regions of inferred too little drag.
Scatter plots of time-averaged zonal dynamics, physics, and momentum nudging tendencies at individual grid points on the lowest four model levels from one run plotted versus corresponding tendencies from the run with twice the nudging time scale are shown in Figure 3. Best fit linear regressions and r 2 values shown in each panel. Nudging tendencies at individual grid points are consistent between runs with τ ndg changed by a factor of two, indicated by the fairly linear relation between the nudging tendencies and the high r 2 (≥0.911) values. Slopes of the linear regressions are significantly larger than one, having values between 1.71 and 1.76. If nudging tendencies were exactly inversely proportional to the nudging time scale, a slope of 2 would result. This is not necessarily expected, however, as reducing the nudging time scale should decrease the differences between the model and target states (and does, shown below). Regardless, the nudging tendencies are roughly proportional to −1 over τ ndg ranging from 3 to 24 hr. The dynamics and physics tendencies are much more consistent when τ ndg is changed by a factor of 2, falling much closer to a 1-to-1 line.
Similar maps and scatter plots of temperature nudging tendencies are shown in Figures 4 and 5. Temperature nudging tendencies have a similar dependence on τ ndg , with tendencies also increasing with decreasing τ ndg . However, this dependence is less strong than that with the momentum nudging tendencies (c.f. Figures 3 and 5a-5c).
If nudging tendencies were to give a robust/consistent estimate of model error of a model in a particular configuration, one might expect that the difference between the model state and the target state (i.e., the numerator in Equation 2) to reduce proportionately to the reduction in nudging time scale. Given that nudging tendencies increase, the difference between the model state and the target state is not being reduced proportionately. The DJF-2008-2012-, level-1-4-averaged momentum nudging tendencies, , in Figure 2 are multiplied by the τ ndg used in each run to illustrate how the time-averaged difference between the model and target state varies in Figure 6. Decreasing the nudging time scale by a 50% does bring the model state closer to the target state, but only by ≈12% (Table 1). For temperature, the nudging term appears a bit more effective, reducing T − T 0 by ≈22% when halving τ ndg (Table 2).
Because nudging does keep the model state close to the target state ( Figure 6) and the DJF-2008-2012-average total wind tendencies are an order of magnitude smaller than the nudging tendencies in all runs (∂ t u ≈ 0, not shown), the momentum budget (e.g., Equation 1) requires a compensation in model tendencies somewhere. This is hinted at by the anti-correlation of the net model tendencies and nudging tendencies in the bottom row of Figure 1, consistent with Bao and Errico (1997) (their Section 4). The changes in dynamics and physics tendencies are plotted versus the changes in nudging tendencies on grid points in Figure 7 for runs with τ ndg changed by a factor of two. Interestingly, this compensation occurs primarily in dynamics tendencies, being particularly clear at the smaller nudging time scales.
These results suggest that, at least when the model configuration is not changed, the physical processes represented by the dynamical core are primarily responsible for compensating these nudging tendencies. In other words, the dynamical core is responsible for fighting the effect of the nudging and reducing the ability of the nudging term to bring the model state closer to the target state. A heuristic theoretical representation of the nudged model system is presented in the following section to speculate on how the dynamical core is fighting the nudging tendencies and why these tendencies are proportional to the inverse nudging time scale.

Time Scale Analysis
Some progress can be made in understanding why nudging tendencies vary with −1 using a simple time scale analysis. Here, the entire host model (i.e., D x (ϕ) + P x (ϕ)) is heuristically represented by a linear relaxation term: where u mod is some zonal wind the model is trying to achieve on some model adjustment timescale, τ mod . A target state for the whole nudged system (u s ) and system relaxation time scale (τ s ) are defined as   With these definitions, the state of the nudged-system can be expressed as a weighted average of the nudging and model adjustment time scales: Correspondingly, the momentum equation can be written as In this simple system, the model is trying to achieve its desired state (i.e., the model is pulling the nudged system state toward its attractor, u mod (x, y, z, t)) while the nudging term is trying to pull the nudged model's state to the target state (u 0 (x, y, z, t), Equation 4). Alternatively, the nudged model state is essentially being relaxed to a timescale-weighted average of the target and model attractor states on the system's adjustment timescale (Equation 8).
Equation 8 implies that, averaged over sufficient time, the time-averaged nudged-model state should just be the time-averaged u s : where 〈.〉 t is a time-average operator. Time-averaged nudging tendencies were presented in the previous section. The following expression for the time-averaged nudging tendencies follows from Equations 2, 7 and 9:  Note. These regressions used time-averaged zonal wind differences differences between the CAM states (u 1 , u 2 ) and the target state (u 0 ) on the lowest four model levels, without the vertical averaging applied in Figure 6, Equation 3. Scatter plots not shown. Slopes indicate how much closer the model zonal wind was kept to the target state by halving the nudging time scale.
This relation suggests that as long as the state the host model is trying to achieve (i.e., its attractor, u mod ) differs from the target state (u 0 ) systematically, there will be non-zero time-averaged nudging tendencies. The magnitude of these nudging tendencies depends on how different the state the host model is trying to achieve and the target state are. The model's adjustment and nudging time scales also play a role. However, if the model's adjustment time scale is sufficiently small relative to the nudging time scale (τ mod ≪ τ ndg ), or it is somehow dictated by the imposed nudging time scale (i.e., τ mod ∝ τ ndg ), then the time-averaged nudging tendencies will be proportional to the inverse of the nudging time scale, consistent with the model result presented above. The model adjustment time scale is further discussed and estimated from the model output next.

Model Adjustment Time Scales
What is this model adjustment time scale and what values might it have? Nudging tendencies likely pull the model away from its desired state, while the model (i.e., the dynamical core in particular) tends to oppose these tendencies as it tries to achieve its desired state. Here, it is speculated that the model relaxation term in Equation 4 represents the host model's adjustment to imbalances produced by dynamical, physical or nudging tendencies. A classical, albeit highly idealized, example of a fluid adjusting from an imbalanced state to a balanced one is geostrophic adjustment (Blumen, 1972). In the geostrophic adjustment problem, the time scale of adjustment is order Rossby number of a geostrophically balanced fluid evolution time scale. In absolute terms, the lower-end of geostrophic adjustment time scales is an inertial period (2π/f, where f is the Coriolis parameter), which depends on latitude, but has values of 12 hr at the poles that increase toward the equator. In reality, and within numerical models, excess momentum associated with a variety of physical imbalances (e.g., deviations from cyclo-geostrophic balance, quasi-geostrophic balance, semi-geostrophic balance, non-linear balance) is shed via generation of, and transport by, atmospheric waves (e.g., gravity waves (GWs), inertia gravity waves, Kelvin waves, (Plougonven & Zhang, 2014)). Outside of the tropics, GWs are emitted from these imbalances, having time scales ranging from the buoyancy period (≈10 min) to the inertial period (≈10 hr). If the time scale of adjustment to the imbalances produced by the nudging tendencies is similar to the time scales of the waves generated by the adjustment process, then τ mod would likely be in the range of 10 min to 10 hr. This range of τ mod (10 min ≲ τ mod ≲ 10 hr) is a bit smaller than the range of τ ndg (3 hr ≤ τ ndg ≤ 24 hr) used here, and so might be consistent with one of the arguments above that nudging tendencies are proportional to −1 because τ mod ≪ τ ndg .
Here, the model relaxation time scales were estimated from the 3-hourly-averaged momentum tendencies by regressing the model tendencies (i.e., D x + P x ) versus the difference between the 3-hourly-averaged nudged-system and model attractor states, u − u mod . This difference is not known because u mod is not known. However, if the differences between the time-varying target state and the model's desired state are much smaller than the time-varying differences of the nudged model system state from the model's desired state or the target state, then u − u mod can be approximated by u − u 0 , which can be easily computed from the nudging tendencies multiplied by the corresponding τ ndg . The model tendencies were regressed against u − u 0 at every grid point on the fourth model level, and the model time scales were then computed from the inverse of the regression slopes. These estimates of τ mod are shown in Figure 8 in both absolute units and relative to the nudging time scale used. Regions color shaded are those where the probability the regression slope is non-zero is greater than 99%. An additional consistency check on this approach is provided by the correlation coefficients between D x + P x and u − u mod (not shown). Correlations are modest but generally significant, with values in the range of −0.5 to −0.75 in parts of the Southern Ocean and Subtropics.
The magnitude of the τ mod estimate is quite variable across the globe (left column of Figure 8). This might not be so surprising, as time scales of geostrophic adjustment and maximum time scales of GWs that can be emitted both decrease with increasing distance from the equator. Zonal asymmetry in τ mod is also quite apparent, both due to significantly smaller τ mod over land versus ocean and zonal variability over just ocean as well. Regions of larger τ mod occur in regions with small nudging tendencies and at low latitudes.
The heuristic analysis above suggested that if τ mod is significantly smaller than τ ndg , or proportional to τ ndg , then the time-averaged nudging tendencies could be expected to be proportional to −1 . Over the Southern Ocean where additional drag is being exerted in all nudging runs, both seem to be true to an extent. The τ mod estimates increase with increasing τ ndg , but less so than expected if τ mod were truly proportional to τ ndg . This is perhaps better seen in the maps of τ mod /τ ndg in Figures 8e and 8f. The ratio τ mod /τ ndg decreases as τ ndg is increased from about 1/2 when τ ndg = 3 hr to about 1/3 when τ ndg = 24 hr.
All this is to say that the model is quite effective at opposing nudging tendencies, in part because the model (the dynamical core in particular) can adjust significantly faster (50%-70%) than the nudging term can pull the nudged model state toward the target state. Interestingly, the model's adjustment time scales also appear to be dictated, to some extent, by the chosen nudging time scale. How the model's dynamical core increasingly opposes the increasing nudging tendencies is interesting. Perhaps the dynamical core makes more efficient use of adjustment mechanisms and instabilities at its disposal to increasingly oppose nudging tendencies' pull away from its desired state? Perhaps additional instability mechanisms are initiated as τ ndg is being reduced? The model's response to the nudging tendencies was not investigated further here.

Can Nudging Tendencies Identify Missing/Erroneous Processes?
Everything presented thus far was based on CAM simulations with the same configuration, but with nudging time scales varied. However, a more practical use of the nudging method might be to change the model configuration (e.g., parameterizations), perhaps as part of a model development or tuning process, and use the nudging tendencies to evaluate whether or not the fast model processes have been improved. While it's clear nudging tendencies cannot be quantitatively interpreted as the net forcing error of the model or all its parameterizations because these tendencies depend strongly on the chosen nudging time scale, nudging tendencies might still point out problem regions qualitatively.
Here, two experiments were conducted to see how nudging tendencies might compensate for Beljaars low-level orographic drag tendencies in runs where that parameterization was turned off. This parameterization was chosen as it exerts very strong forces on the low-level flow and has an easily recognizable spatial distribution (i.e., it mainly acts over significant mountain ranges). In the first experiment, two runs were conducted, both being nudged to ERA5 with τ ndg = 6 hr, with the Beljaars parameterization turned off in one of the runs. In the second experiment, a free-running CAM6 run was integrated for five years and three months, initialized from ERA5 on 1 Sept 2008. Then, a second CAM configuration without the Beljaars parameterization was initialized by the free-running run every 1 Dec from 2008 to 2012 and run for three months, being nudged to the free-running run with a τ ndg = 6 hr as well.
The absolute Beljaars tendencies projected along the wind and averaged over the lowest four model levels are shown in the left column of Figure 9. The absolute changes in nudging tendencies between the run with and without the Beljaars parameterization, with both runs nudged to ERA5, are shown in Figure 9b. The nudging tendencies in the run nudged to the free-running run without the Beljaars tendencies are shown in Figure 9d. Encouragingly, when this parameterization is removed, nudging tendencies do act to replace these parameterized tendencies (compare right and left columns of Figure 9) in both experiments. However, the low-level drag added by the nudging to replace this process is both more spatially spread out and only ≈16.5% of the tendencies by this process in the run with it enabled, from linearly regressing the right column against the left column of Figure 9 (Regression slopes were 0.16 and 0.17. Both experiments had correlation coefficients of r = 0.84. Not shown.). While the results of both of these experiments are very consistent, there are broad regions of small, but non-zero nudging tendencies away from regions with orography in the second experiment. These features are presumably associated with the key difference between the experiments, that is, the lack of nudging in the control of the second experiment, though, these features were not investigated further here. These results further suggest that, while nudging tendencies can useful in identifying model errors and/or missing processes, these tendencies should only be used qualitatively.

Summary
The primary result of this article is that nudging tendencies that result from nudging a model to a target state are proportional to the inverse of the nudging time scale chosen (Figs. 2, 3a-c). As momentum nudging tendencies increase, they are increasingly compensated by changes in tendencies by the dynamical core (Figure 7, top row), presumably to adjust to imbalances introduced by the nudging. A heuristic time scale analysis suggests that if the model's imbalance adjustment time scale is significantly smaller than, or proportional to, the chosen nudging time scales, this result should be expected as long as the target state to which the model is being nudged is different from the state the model is trying to achieve, regardless of the type of target state (e.g., a native analysis, a non-native analysis, some other model entirely that has not assimilated data). The results presented here suggest a model's imbalance adjustment time scale might be both smaller than, and somewhat proportional to, the chosen nudging time scale (Figure 8).
The consequence of this result is quite important for the interpretation of nudging tendencies. If nudging tendencies were to be interpreted as total model or physics errors, then the errors so inferred would depend on the nudging time scale chosen. Still, systematic nudging tendencies represent a tendency of the model to push its state away from the target state. If the target state is considered "correct," then these tendencies still contain some information on model error, even if only qualitatively. However, nudging tendencies cannot be used to, for example, quantitatively identify forcing error profiles by a single parameterization known to dominate forcing in a particular profile.
The relevance of this result is significant as well. Nudging is a ubiquitous capability of numerical weather and climate models and is used widely in a variety of applications. While few have made use of the nudging tendencies themselves, an emerging application is to use methods to predict nudging tendencies of a model given its state in order to improve its predictive skill and reduce long-term biases. Watt-Meyer et al. (2021) and Bretherton et al. (2021) used machine learning to do this, with some success, though they differed in the variables nudged and focused on. The results here do not invalidate such results, but suggest that they may depend not only on the model and resolution used, but also on the nudging time scale chosen.
The nudging method of data assimilation and model error quantification is likely the simplest in a spectrum of such methods. The results here are not expected to have any relevance to the initial tendency method or increment tendency methods of model error quantification (e.g., Klinker and Sardeshmukh (1992); Rodwell and Palmer (2007); Cavallo et al. (2016)). However, there may be some relevance to the incremental analysis update (IAU) method (Bloom et al., 1996;Gelaro et al., 2017;Rienecker et al., 2011;Takacs et al., 2018), which applies analysis increment tendencies, held constant over a period equal to the spacing between guiding analyses (e.g., 6 hours), in governing equations to "guide" a model's state along an analyzed trajectory. While there are similarities between the IAU and nudging methods, it is unclear at this point if a similar dependence of IAU tendencies on the IAU time scale might exist. Figure 9. The DJF-2008-2012-averaged Beljaars absolute tendency in the default CAM6 run nudged to ERA5 is color shaded in (a). The absolute difference in nudging tendencies between ERA5-nudged-runs with (CAM6) and without (NoBLJ) the Beljaars parameterization are color shaded in (b). Beljaars tendencies in a free-running CAM6 run are color shaded in (c). The nudging tendencies in a run without Beljaars nudged to the free-running run are color shaded in (d). Horizontal wind tendencies on the lowest four model levels were projected along the horizontal wind vector and then vertically averaged weighted by density. Both runs were nudged using τ ndg = 6 hr. Comparison of right and left columns give an indication of how nudging tendencies might indicate deficient or missing physics.

Data Availability Statement
Monthly mean output presented here along with the scripts used to setup CAM6 have been archived and are accessible via the Zenodo repository Kruse et al. (2022). Model output at 3-hourly resolution used to estimate model adjustment time scales may be made available via special request to ckruse@nwra.com. ERA5 reanalyses were regridded onto the CAM grid used here via the Betacast software package (Zarzycki, 2022;Zarzycki & Jablonowski, 2015).