## 1. Introduction

Errors in modeled land surface properties that affect fluxes at the land–atmosphere interface can affect short-range, mesoscale predictability in the atmosphere in two important ways. It will quickly introduce an error into the atmospheric surface layer, and consequently into horizontal and vertical pressure gradients. Ensuing low-level wind errors in turn may spread via processes such as boundary layer impingement on the free atmosphere or convective triggering. Concurrently, error near the surface will be continually introduced by the error in the lower boundary specification.

The aim of this study is to characterize mesoscale, short-range (0–48 h) uncertainty in PBL winds as it responds to uncertainty in land surface specifications. The research presented here is exploratory; it is intended to guide further work focusing on particular aspects of the response at scales characteristic of operational forecasting systems in the near future. Choosing soil moisture as the land surface state variable subject to uncertainty, the specific goals of this work are threefold: 1) to quantify spatial and temporal scales of near-surface wind predictability (or lack of predictability) as it relates to small-amplitude error in soil moisture specification, 2) to determine whether atmospheric conditions or perturbation scale and amplitude are the leading factors in PBL wind error growth, and 3) to measure nonlinearity in the error growth during different conditions.

Observing mesoscale predictability in the atmosphere is extremely difficult, if not impossible. The Weather Research and Forecast (WRF) model coupled to the Noah land surface model (LSM; Ek et al. 2003; Chen et al. 2007) serves as an alternative laboratory, and predictability is assessed through a series of soil moisture perturbation experiments in which the model is considered perfect. We measure error growth intrinsic to the WRF/Noah system and interpret it as a lower bound on total error growth in a prediction system. Methods and diagnostics are inspired by the approach used in several earlier predictability studies (Errico and Baumhefner 1987; Tribbia and Baumhefner 1988, 2004), none of which address mesoscales, low-level winds, or soil moisture. The experiment design allows quantification of scale-dependent predictability and error growth in the atmosphere as it responds to soil moisture uncertainty.

Uncertainty prescribed to soil moisture in these experiments is not intended to reproduce local soil moisture uncertainty and is in some ways unrealistic, but it is general and provides the lower-boundary condition uncertainty needed to examine the atmospheric response. Niyogi et al. (1999) showed that the response to uncertainty in a single land surface parameter is dynamic and depends on the values of other parameters. Only soil moisture is perturbed here to avoid that complexity. To avoid results dependent on details of the soil moisture perturbations, we take care to make them smaller than reported soil moisture analysis errors. They are also random in the perturbed scales, and although a component of soil moisture uncertainty might appear random we make no attempt to ensure that the structure of the randomness adheres to any known uncertainty structures. As results will show, the details of the perturbations do not determine the main conclusions.

The sensitivity of moist atmospheric processes to soil conditions has been demonstrated in numerous studies, which collectively show that clouds and precipitation are sensitive to soil moisture at a wide range of spatial and temporal scales. Pielke (2001) reviewed work on this subject and provides extensive references. Other examples include Holt et al. (2006), who analyzed a case study of simulated deep convection during the International H_{2}O Project 2002 (IHOP_2002; Weckworth et al. 2004). They found that the Noah LSM improved simulations of deep convection compared to a simpler LSM based on force-restore. Adding a more sophisticated treatment of evapotranspiration further improved simulations. Cheng and Cotton (2004) documented how the evolution of a mesoscale convective system (MCS) simulated with 4-km horizontal grid spacing changed in response to systematic variations of soil moisture heterogeneity. They interpreted their results to suggest that soil moisture analyses limited to scales greater than 40 km may be sufficient for simulating MCSs at much finer scale and that the locations of soil moisture anomalies largely determine the locations of deep convection.

The effect of soil moisture uncertainty on mesoscale circulations and PBL structure can also be deduced from studies of their physical link. Mahfouf et al. (1987) used idealized two-dimensional (vertical plane) simulations to show that soil moisture variability at scales near 100 km can produce significant circulations. Using a calibrated evapotranspiration model, Pinty et al. (1989) showed sensitivity of two-dimensional circulations to the vegetation contrasts at scales near 100 km; their results suggest that sophisticated evapotranspiration schemes require accurate soil moisture specification to provide accurate forcing on the atmosphere. Górska et al. (2008) used aircraft observations during an IHOP_2002 day lacking nearby deep convection to find that turbulent boundary layer fluxes of moisture 70 m above ground corresponded more with the normalized differential vegetation index (NDVI) than surface moisture fluxes. They point out that heterogeneity observed in the NDVI was not well represented by the land surface state estimate used to provide surface flux information for their corresponding large-eddy simulation study.

Using a single-column model (SCM), Zhang and Anthes (1982) documented the sensitivity of PBL profiles to land surface properties including moisture availability. Also using an SCM, Alapaty et al. (1997) found that uncertainty in soil texture and stomatal resistance can have a significant impact on simulated PBL turbulent heat fluxes and mean thermodynamic properties, but wind errors are insensitive. Niyogi et al. (1999) found that moderate uncertainties in multiple land surface properties can compensate for each other and lead to small PBL prediction errors when surface conditions are not near their extreme values. Conversely, under conditions near the wilting or saturation points the prediction errors can be large because, for example, uncertainty in specification of wilting or saturation points can interact synergistically with uncertainty in other surface parameters. Because these studies used a single-column model and restricted attention to the PBL, nonlinear advection in the atmosphere and interactions with moist processes were not considered.

The present study builds on those cited above by using a three-dimensional mesoscale model with moist physics as an atmospheric proxy. We focus on spatially correlated small-amplitude soil moisture uncertainty at scales up to 64 km, assume that the other land surface parameters are perfectly specified, and quantify spatial and temporal scales of intrinsic error growth. The experiments, and an objective scale-dependent perturbation strategy that preserves the spatial covariance of the soil moisture field, are described in the next section. Section 2 also shows that the perturbation amplitude is small. Predictability is assessed in section 3, with further interpretation in section 4. Section 5 contains a summary of the key results.

## 2. Experiment design

### a. Study periods and model implementation

Two periods from IHOP_2002 provide contrasting analyses: 27–29 May and 6–8 June 2002. Both are characterized by weak synoptic gradients over the U.S. Great Plains, but the occurrence of deep convection is more widespread during 27–29 May, indicating static instability. Figure 1 shows deep convection (noted by arrows) initiated near the dryline on 27–28 May, but a lack of deep clouds over the Southern Great Plains at midday during 6–7 June. Weak upper-level troughs were present during both periods and were oriented southwest to northeast through the region. Upper-level horizontal pressure variations were slightly greater during 27–28 May. This study does not address the role of upper-level forcing in the onset of deep convection; rather, it focuses on the role of deep convection in propagating PBL wind uncertainty.

The Advanced Research WRF (ARW) version 2.2 on two-way nested domains with 12- and 4-km grid spacing, summarized in Table 1, is the experiment platform. The WRF is initialized independently at 1800 LST 26 May and 5 June 2002. Atmospheric initial conditions and lateral boundary conditions are taken from the Eta model analyses provided every 12 h with horizontal grid spacing of approximately 40 km. The sixth-order numerical diffusion scheme described in Knievel et al. (2007) and Xue (2000) is applied to ensure that spurious 2Δ*X* waves do not form during weak wind conditions in the convective PBL. All analysis is performed on the inner domain with Δ*X* = 4 km; the outer domain is used mainly to buffer the inner domain from the Eta model–derived lateral boundary conditions.

Specific choices of parameterization schemes can potentially change the number and behavior of sensitive modes in a WRF simulation and lead to different estimates of predictability. But in general any possible WRF configuration will have less variability than the atmosphere. When characterizing sensitivity, the difference between the model and the atmosphere is likely to be much greater than the differences among different choices of physics schemes. Thus, the specific suite of schemes is not critical as long as they are reasonable for mesoscale problems. First-moment systematic errors such as a bias in 10-m winds are also of secondary concern here because the analysis focuses on departures from unperturbed simulation rather than the skill of any simulation.

The Yonsei University (YSU) PBL scheme (Troen and Mahrt 1986; Hong and Pan 1996; Noh et al. 2003; Hong et al. 2006) parameterizes vertical turbulent mixing with a first-order closure. It is a *K*-profile method with a countergradient term and explicitly estimates the entrainment proportional to surface buoyancy flux. A parameterization for deep convection is not used on the 4-km grid, and the Kain–Fritsch (Kain and Fritsch 1990; Kain 2004) scheme is used on the 12-km grid. Radiation is handled with the Rapid Radiative Transfer Model (RRTM; Mlawer et al. 1997) and the Dudhia (1989) longwave and shortwave schemes, respectively.

For microphysics we adopt the simple ice scheme described in Hong et al. (2004), which is referred to as the WRF single-moment three-class (WSM3) scheme. It is a relatively simple scheme, but when comparing several different simulations it produces the best spatial distribution of 24-h precipitation compared to the National Centers for Environmental Prediction (NCEP) stage IV precipitation analysis for the study period. A histogram of the precipitation from the resulting WRF simulation and the verifying stage IV analysis, for the period 0600 LST 27 May to 0600 LST 28 May 2002, is shown in Fig. 2. The stage IV analysis is on a grid with 4-km spacing, and the fields are interpolated to the 4-km WRF grid without much loss of information. The WRF did not simulate the greatest precipitation totals in the analysis, which is a deficiency that we hope will not significantly change the number of grid cells with saturated soil. Chen et al. (2007) found a reasonable agreement between stage IV analyses and rain gauge data at most IHOP_2002 observing locations in the Southern Great Plains during May–June 2002, but with some instances of large over- or underestimation by the stage IV analyses.

The Noah LSM available with the ARW version 2.2, described by Chen et al. (2007) and references therein, handles land surface modeling. Spun-up states from the National Center for Atmospheric Research (NCAR) High-Resolution Land Data Assimilation System (HRLDAS) provide the initial soil (temperature and volumetric moisture) analyses for 1800 LST 26 May and 5 June 2002. Chen et al. (2007) give details of the HRLDAS and this particular implementation. Initial conditions for the Noah LSM are prescribed for 1 January 2001 from an NCEP Eta model data assimilation system (EDAS; Rogers et al. 1999) analysis. Forcing for LSM integration through 1 July 2002 includes the following: observed precipitation from the NCEP stage IV analyses, downward solar radiation derived from Geostationary Operational Environmental Satellite (GOES) observations, downward longwave radiation, 10-m wind speed, 2-m temperature, and 2-m specific humidity from the 3-hourly EDAS analyses. Chen et al. (2007) verify the analyses favorably against observations during the IHOP_2002 period. In our WRF simulations we use the same LSM vertical grid as Chen et al. (2007), with layer thicknesses of 0.1, 0.3, 0.6, and 1.0 m from top to bottom (i.e., midlayer depths of 0.05, 0.25, 0.7, and 1.5 m) on the 4-km grid.

LeMone et al. (2008) evaluated HRLDAS surface temperatures and fluxes against a 50-km IHOP_2002 flight track on 29 May 2002. They found that manual modifications to both the soil moisture and the Zilitinkevich (1995) coefficient (*C*) for heat and water vapor “roughness” were needed for the HRLDAS to improve mean gradients of surface sensible and latent heat fluxes along the flight track. LeMone et al. (2008) also found that the sensitivity to *C* was much greater than sensitivity to the soil moisture. The present experiments use the default *C* = 0.1, and if the results of LeMone et al. (2008) are general to the entire domain then the horizontal variability at scales below 50 km may be underestimated in the control simulation. Assuming that the HRLDAS states used in these experiments are lacking in finescale soil moisture variability (e.g., because of inherent truncation errors in the prescribed precipitation), and that the Noah with *C* = 0.1 will lack finescale variability in surface heat fluxes, we expect that predictability estimates will be conservative (i.e., true predictability time scales will be shorter).

### b. Perturbation strategy

Experiments are constructed to quantify the response to soil moisture perturbations during both day and night. As summarized in Fig. 3, a control simulation is initialized at 1800 LST and integrated for 48 h. We introduce pairs of perturbations on the 4-km domain at two times: 0000 and 1200 LST.

We assume that the spatial structure of a given soil moisture estimate is realistic within the limitations of the observation network, land surface model, and grid spacing used to derive it. From this we construct perturbations to represent random, spatially correlated errors with stationary spatial covariance. We adopt a generic perturbation approach that can potentially elicit an atmospheric response, with specific perturbation properties defined by the instantaneous volumetric soil moisture (hereafter simply “soil moisture”) state in the control WRF simulation. The method does not attempt to reproduce the characteristics of actual soil moisture uncertainty, which we expect varies locally and regionally with the properties of the soil, vegetation, and background moisture itself. Rather, we impose small-amplitude perturbations that can plausibly represent part of the uncertainty in most places.

*k*. Two-dimensional Fourier decompositions on a limited-area domain can be computed similarly to those in Errico (1985). A perturbed soil moisture field is constructed by recombining the Fourier coefficients after perturbing them with random rotations asThe perturbed coefficient

_{c}*iϕ*,

*ϕ*∼

*π*). A uniform distribution for the rotations is appropriate here to simulate a complete loss of information at scales below

*k*; rotations drawn from a normal distribution would increase the likelihood that a perturbed state is close to an analyzed state. Because the perturbations are small, we apply the same perturbations to all four levels in the soil model and avoid explicitly perturbing vertical gradients in the LSM. Another effect is to give all transpiring plants equal access to the perturbed soil moisture regardless of root depth. Perturbations symmetric about zero are constructed by simply reversing the sign of a single 2D perturbation field (i.e., equal-magnitude but opposite perturbations are computed at each model grid point).

_{c}Here *k _{c}* corresponds to a wavelength

*λ*, which we choose to be a multiple of the grid spacing. Because guidance on a generic uncertainty length scale in soil moisture analyses is lacking, experiments with

_{c}*λ*= 16 km and

_{c}*λ*= 64 km are analyzed. Skamarock (2004) averaged 1D wind spectra over a 3D domain and found that the WRF-ARW dynamics show a diffusive range diagnosed by a downturn in the spectra at scales below approximately 7Δ

_{c}*X*(here 28 km). Knievel et al. (2007) found a diffusive range below 6Δ

*X*in the PBL when imposing sixth-order numerical diffusion. Cutoff scales of 16 and 64 km thus represent different regimes with respect to numerical diffusion in the WRF. Note too that a sloped soil moisture analysis spectrum results in perturbations imposed at 64 km that are an order of magnitude greater than 16-km perturbations.

The perturbation method follows earlier predictability experiments that focused on atmospheric initial condition error (e.g., Errico and Baumhefner 1987; Mullen and Baumhefner 1989; Tribbia and Baumhefner 2004). With only minor variations, they rotated all Fourier coefficients but then scaled those greater than approximately *k _{c}* = 30 so that the uncertainty spectrum was white with amplitude equal to the analysis variance amplitude at

*k*= 30. Those perturbations were intended to simulate analysis error as initially quantified by Daley and Mayer (1986).

_{c}Although random perturbations do not project strongly on the fastest-growing modes in atmospheric models, and error amplitude really varies spatially, the error-growth characterizations from that work remain salient. The perturbations here are most similar in nature to the small-scale perturbations applied in Tribbia and Baumhefner (2004), where uncertainty at scales greater than *k _{c}* was made very small, but here it is zero. Those experiments examined growth from atmospheric perturbations at relatively small scales in a global model that did not resolve mesoscale dynamics. Here mesoscale perturbations and responses are the primary interest, and

*k*≈ 350 (64-km perturbations) or 1250 (16-km perturbations).

_{c}### c. Perturbations in physical space

Sensitivity to volumetric soil moisture *w _{υ}* must be considered in the context of the plant wilting point

*w*

_{pwp}and the field capacity

*w*

_{fc}, which are properties of the soil type. Specifically,

*w*

_{fc}is the maximum moisture the soil can hold against the earth’s gravitational acceleration. Below

*w*

_{pwp}plants stop transpiring because sufficient moisture is unavailable.

Plants can transpire where *w*_{pwp} < *w _{υ}* <

*w*

_{fc}; surface sensible and latent heat fluxes are more sensitive to soil moisture perturbation. After perturbation at 1200 LST 27 May 2002, 99.90% and 99.85% of the points in the control and perturbed soil moisture fields, respectively, are inside the sensitive range. Transpiration is possible and perturbations can directly modify the flux at most grid points. In the other case, 6 June, the soil is much dryer and only about 13% of the points are within the sensitive range. Evaporation is then the primary mechanism for latent heat flux. It is less efficient than transpiration, and the perturbations will have much less direct effect on the fluxes.

Perturbations imposed in these experiments are small compared to both typical soil moisture analysis errors and perturbations used in other experiments. Figure 4 shows the perturbations Δ*w _{υ}* =

*w*(perturbed) −

_{υ}*w*(control) at 1200 LST 27 May 2002. Values of

_{υ}*w*are

_{υ}*O*(0.2 m

^{3}m

^{−3}) on 27 May; Fig. 4 shows Δ

*w*≤ 0.04 m

_{υ}^{3}m

^{−3}. The standard deviation of the distribution in Fig. 4 is 0.01 m

^{3}m

^{−3}so that 68% of the grid points have Δ

*w*≤ 0.01 m

_{υ}^{3}m

^{−3}. Chen et al. (2007) verified

*w*, soil moisture, and temperature against Oklahoma Mesonet measurements, finding errors ranging between 0 and 0.04 m

_{υ}^{3}m

^{−3}at both 5- and 25-cm depths. Their results are consistent with Drusch and Viterbo (2007), who found errors of 0–0.04 m

^{3}m

^{−3}reported at 5-cm and 1-m depths in analyzed soil moisture from the land data assimilation system used at the European Centre for Medium-Range Weather Forecasts (ECMWF). For further context, Alapaty et al. (1997) and Niyogi et al. (1999) used perturbations up to 0.09 m

^{3}m

^{−3}in their single-column model experiments.

## 3. Results

### a. General description of response

Here and later, the difference of a variable *ψ* between the two perturbed simulations is the basis for quantitative analysis: *D _{ψ}* =

*ψ*

_{+}−

*ψ*

_{−}, where subscript + and − indicate simulations starting from perturbations added and subtracted to the background, respectively. The spatial variance of

*D*(denoted

_{ψ}*V*) or its square root (

_{D}*σ*) are used as summary metrics. One way to measure predictability and the propagation of uncertainty is to characterize

_{D}*V*as function of time.

_{D}Precipitation sensitivity to soil moisture perturbations highlights differences between the two days studied here and helps interpretation later. Figure 5 shows one-hourly accumulated precipitation domain sums (bars) from the control simulations, as well as *σ _{D}* following perturbations applied 1200 LST. Domain-sum precipitation is two orders of magnitude greater on 27–28 May than 6–7 June; rain falls more on the soil that already contains more moisture. Precipitation prior to 1800 LST 6 June (Fig. 5b) moistens the soil so that the fraction of the domain where sufficient moisture is available for transpiration rises from approximately 7% to 13%. It remains at about 13% through 1800 LST 7 June. More precipitation occurs with greater precipitation

*V*.

_{D}*D*, the kinetic energy (KE) of the difference between 10-m vector winds (

_{ψ}*U*,

*V*) in the two perturbed simulations isIn the context of these perfect-model experiments,

*E*is an error kinetic energy.

It is instructive to first see how local and regional variations of KE and *E* contribute to the metrics computed later. Band-passed *E* fields identify atmospheric conditions responsible for upscale error transfer. Here we apply a Gaussian bandpass filter with bands centered on 64 and 600 km; *σ* = 120 km in the Gaussian function produces a smooth filter and minimizes scale leakage when inverting the filtered Fourier coefficients. Figure 6 shows results valid 24 h after the 1200 LST 27 May 64-km perturbation experiment. Results from the 16-km experiment are qualitatively similar.

Figure 6 shows upscale error transfer tied to deep convection. The greatest error kinetic energy in both scale bands is near the most convectively active regions, along the line marked A–B. Figures 6a and 6b show that the WRF simulations broadly capture the key features of this event (cf. Fig. 1b), and Figs. 6b and 6d show local and regional error energy *E* approaching or greater than the kinetic energy itself.

Regions where the *E* field (Fig. 6, right panels) exceeds the KE field (Fig. 6, left panels) indicate a local loss of predictability. At 24 h a local loss of predictability is apparent in the smallest scales near the deep convection (Fig. 6a), with *E* > KE to the east of point A and southwest of point B in Fig. 6b. Figure 6d shows regional loss of predictability at scales much larger than the perturbations.

The greatest values of *E* follow propagating deep convection. Figure 7 shows two 6-h periods of error at the 600-km scale, separated by 12 h. Large-scale errors appear to follow the regions of deep convection, and although the errors grow upscale they do not propagate spatially far from their origins. Figures 7a and 7b show sharp arc-shaped error structures. Examination of more satellite images (not shown) confirms that these regions broadly agree with regions of deep convection. Also, squall lines are not visible in any of the images. By deduction, then, we interpret these arcs as thunderstorm outflow boundaries.

These examples illustrate that deep convection plays a role in local upscale error transfer. It further shows that *E* does not exceed KE within much of the domain. Domain-wide metrics computed in the remainder of this paper mask the local loss of predictability because isolated regions of predictability loss are averaged with large areas of high predictability.

### b. Error growth

Growth of the spatial variance of *E* (*V _{E}*) can be interpreted as error growth and shows the time-dependent response to perturbation. Note that

*V*is normalized by the spatial variance of the control KE (

_{E}*V*) valid at the same time to give a relative error growth;

_{C}*V*/

_{E}*V*≥ 1 generally indicates useless predictability (i.e., no skill in a prediction with a perfect model) and

_{C}*V*/

_{E}*V*= 2 indicates error saturation.

_{C}The *V _{E}* /

*V*curves for all experiments are shown in Fig. 8, where the top (bottom) panels show results from simulations with soil moisture perturbations applied at

_{C}*λ*≤ 16 (

_{c}*λ*≤ 64) km. Figures 8b and 8d show the same results as Figs. 8a,c but are plotted on a semilogarithmic axis to highlight the results from 6 June. We restrict attention to times after 1200 LST on the first day, coincident with application of the 1200 LST perturbations, for easier comparison to later results.

_{c}Maximum error level and peak-error timing differences between the two days are the most obvious features. Maxima during 27–28 May range from 0.2 to 0.5, but maxima during 6–7 June range from 10^{−4} to 10^{−2}. Initial peaks during 6 June are at 2100 LST or earlier, but those on 27 May appear after 2100 LST. The relative amplitude of the oscillation is greater during 6–7 June, where an early-morning minimum is three orders of magnitude smaller than the maxima. Oscillation magnitude during 27–28 May is smaller at approximately 0.2.

Perturbation scale and magnitude are secondary factors on 27 May but can cause large differences in initial error growth on 6 June. Varying *λ _{c}* from 16 to 64 km during 27 May leads to differences less than 0.1 in the first error peaks near 0000 LST. During 6 June, the 1200 LST perturbations lead to differences of up to two orders of magnitude in the first peaks near 1800 LST (dashed–dotted curves), although differences following the 0000 LST perturbations appear minor (dotted curves).

Initial growth followed by an oscillation, and little difference between the responses to the two perturbation scales on one day, suggests that the state of the atmosphere is the leading factor in determining the maximum wind response. Whether or not the plants are transpiring, error growth is minimal between 0600 and 1200 LST, and soil moisture uncertainty by itself does not appear capable of driving large error growth in the winds. During both days the oscillation is shorter than the length of the day would suggest, and the peaks are closer to the precipitation peaks in Fig. 5. The presence of deep convection determines whether error growth can be large and whether the perturbation scales are important; it occurs in the afternoon of 27 May and continues into the evening before reinitiating during the late morning of 28 May (Figs. 5 –7). On the dry days (6–7 June) the magnitude of wind error is an order of magnitude smaller, and the oscillations are greater in relative amplitude. More predictability is maintained, and error energy decreases as precipitation and surface fluxes decrease in the evening.

Figure 8 shows that further examination of the 0000 LST perturbations will generally not add insight. Error growth occurs after sunrise on the subsequent day. Henceforth we restrict our attention mainly to perturbations at 1200 LST.

### c. Dependence on spatial scales

Results presented in Figs. 6 –8 motivate analysis of scale-dependent contributions to error growth. A 2D Fourier decomposition of *E* and control KE enables separation of normalized error growth into spectral bands. Three bands represent separately the grid and perturbation scales (0–80 km), inertial wave and deep convection scales (140–260 km), and large but subsynoptic scales (380–780 km). Synoptic and planetary scales are primarily controlled by the time-dependent lateral boundary conditions, which are invariant between the control and perturbed simulations.

Results in Fig. 9 show that relative error growth is greatest in the smallest (near grid-scale) band, corresponding to the perturbation scale, but the perturbations can also transfer to larger scales. Note the different ordinate scales. As expected from Fig. 8, values for 27–28 May (Figs. 9a,b) are greater than those for 6–7 June (Figs. 9c,d). Upscale error transfer during 27–28 May leads to normalized error in the larger-scale bands of about half the magnitude of error in the perturbation scale. During the second day of 6–7 June, some upscale error transfer is present, but the normalized error in the middle-scale (large-scale) band is about 0.4 (0.1) of the perturbation-scale band. It appears that the atmosphere is capable of upscale error energy transfer on both days, but the rates of total error growth and upscale transfer are greater on 27–28 May when the atmosphere is statically unstable.

Cross-spectral analysis quantifies the scale-dependent linear relationship between error energy in the surface sensible heat flux and the winds. The difference *D _{H}* =

*H*

_{+}−

*H*

_{−}, where

*H*is the sensible heat flux, is analyzed with

*E*. Given 2D Fourier coefficients

*E*and

*D*, the cross spectrum between

_{H}*E*and

*D*is defined as

_{H}*A*=

Although *A* is not strictly the covariance between *E* and *D _{H}*, peaks in

*A*indicate a strong covariance between

*E*and

*D*at a particular wavelength. The key drawback is that

_{H}*A*is not normalized, and it can be large in magnitude when either

*V*or

_{E}*V*is large at a particular scale and the correlation between

_{DH}*E*and

*D*is strong (cf. Stull 1998).

_{H}Amplitude spectra show that the soil moisture perturbation scales persistently affect wind-error scales during 6–7 June (deep convection absent) but not during 27–28 May (deep convection present). Figure 10 shows amplitude spectra valid 1 h (circles), 2 h (diamonds), and 24 h (squares) after perturbations are applied at 1200 LST. Solid curves result from perturbations applied at and below 16 km, and dashed curves result from perturbations applied at and below 64 km. On both days, peaks in the 1-h spectra show the perturbation scales of 16 (solid circle) and 64 km (dashed circle). Amplitude at and below the 16-km perturbation scale at 1300 LST 27 May (Fig. 10a) is notably greater than at 1300 LST 6 June (Fig. 10b) because small-scale wind errors rapidly develop when evapotranspiration is active on 27 May. Between 1300 and 1400 LST on both days the amplitude at scales larger than the perturbations also grows rapidly; perturbation-scale peaks are still visible but are less prominent by 1400 LST (diamonds).

During the following 22 h differences between the two cases become clear. Perturbation-scale peaks are gone by 1200 LST 28 May (squares in Fig. 10a), and the amplitude spectra resulting from the 16- and 64-km perturbation experiments are largely similar (squares), showing that atmospheric error growth dominates. However, at 1200 LST 7 June (squares in Fig. 10b) the perturbation scales are still marked by peaks in the amplitude spectra. Amplitudes at all scales are also smaller, and the large-scale amplitude spectra are flatter, indicating a lack of large-scale error growth.

Comparing soil moisture difference (*D _{wυ}*) spectral densities (Fig. 11) to the amplitude spectra (Fig. 10) shows that by 24 h, uncertainty in the fluxes and winds are not controlled by the evolution of the soil moisture perturbations during 27–28 May. Whether or not deep convection is present, an imprint of the soil moisture perturbations remains throughout the simulations. Convective cells produce precipitation over regions larger than their instantaneous size and can contribute to large-scale soil moisture uncertainty here. Wind uncertainty caused by the perturbed evolution of the deep convection produces some large-scale uncertainty in the soil moisture (squares in Fig. 11a) through variable drying. Neither drying nor precipitation is enough to explain large-scale growth in the amplitude spectra. Large-scale soil moisture variances remain smaller than at perturbation scales, clarifying that deep convection and wind uncertainty determine the large-scale response in the amplitude spectra (Fig. 10).

Results presented in this section lead to the conclusion that the soil moisture uncertainty scale can determine the PBL wind uncertainty scale on days when the atmosphere is statically stable and deep convection is sparse. On those days, improved soil moisture analyses and LSMs should improve numerical predictions. Conversely, conditions supporting upscale error transfer can quickly render the specific characteristics of soil moisture uncertainty irrelevant with respect to PBL winds.

## 4. Nonlinearity

A growing response to small perturbations is one prerequisite for deterministic chaos; it implies the existence of a limit to predictability. The results from section 3 suggest periods of nonlinear error growth (Figs. 8 and 9), and here we formally quantify it using finite-size Lyapunov exponents (FSLEs).

The FSLE spectrum (Aurell et al. 1997) is an extension of the maximum Lyapunov exponent to finite-sized perturbations. Infinitesimal perturbations are difficult to estimate from discrete systems with spatial and temporal scale truncation, such as an NWP model, and in any case may not provide the best description of predictability times (Boffetta et al. 1998).

*λ*(

*δ*

_{0}), is a function of a finite perturbation magnitude

*δ*

_{0}, where the perturbation here is a nonzero

*E*and may have been caused by soil moisture perturbations. Analogous to the maximum Lyapunov exponent,where the angle brackets are a sample mean and

*δ*=

*rδ*

_{0}is the perturbation size after time

*T*. Following convention,

_{r}*r*= 2 to quantify doubling times. Values for

*δ*

_{0}are selected to provide reasonable sample sizes for estimating

*λ*(

*δ*

_{0}).

Samples for computing means are formed by searching the first 24 h of the *E* fields for points where *E* is within bins centered on values of *δ*_{0}. The time *T _{r}* for that grid point is then the minimum time measured for

*E*to reach 2

*δ*

_{0}. Results are excluded from the sample if a perturbation fails to double. Although this sampling strategy is not optimal in terms of independence, the subsample at each time is frequently distributed across many regions in the grid, and we thus expect a reasonable level of independence. Here we do not compute an FSLE for a bin with a sample size less than 100.

FSLE spectra typically show three dynamical regimes, corresponding to fast response to small-magnitude perturbations, moderate response to moderate perturbations, and slow response to large-magnitude perturbations (e.g., Boffetta et al. 1998; Tao and Barros 2008). The smallest perturbation measured corresponds to the smallest wind error energy after 6 min of simulation (the first output time analyzed) and is *O*(1.3^{−12}) m^{2} s^{−2}.

Results (Fig. 12) show the possibility of strong nonlinearity in the wind response to soil moisture within both the convectively active (solid curves) and more stable (dashed curves) simulations. A nonlinear response is possible whether or not the vegetation is actively transpiring. To contrast different times and states, Figs 12a,c,e are from the 64-km 1200 LST experiments, and Figs. 12b,d,f are from the 16-km 0000 LST perturbation experiments (note the different plot scales). Figure 12a shows positive *λ*(*δ*_{0}) for a wide range of *δ*_{0}.

The slope in the curves toward the left side of the plots suggests that the response to the smallest viable perturbations are in the moderate-response regime, which is evident for perturbations between 10^{−12} and 10^{−7} m^{2} s^{−2}. These result in perturbation doubling times much less than an hour (Fig. 12b), coincident with the (parameterized) PBL eddy turnover time. Perturbations between 10^{−7} and 10^{−2} m^{2} s^{−2} result in perturbations between tens of minutes and two hours and can be considered a slow response because of the flatness and small positive values of the FSLE curves in that range. Even slower responses, apparently approaching the diurnal cycle, are evident at perturbations between 0.1 and 1.0 m^{2} s^{−2}.

Large-magnitude perturbation growth in *E* is robust with respect to the time of day and to the scale and magnitude of the soil moisture perturbations, but small-perturbation growth depends on the experiment. Considering only the 1200 LST 64-km perturbations (Figs. 12a,c,e), the moist day with deep convection (solid) can be characterized by slower-growing small perturbations, and faster-growing large perturbations, than on the dry day (dashed). Because perturbations do not grow as rapidly or over as much of the domain on the dry day, it yields a larger sample of small-magnitude perturbations (Fig. 12e).

Smaller scale and magnitude (16 km) soil moisture perturbations applied at 0000 LST (Figs. 12b,d,f) show the same results for larger *E* perturbations, but the growth of small perturbations does not vary between days. Small *E* near 1200 LST 27 May rapidly grow to larger magnitude, leaving small sample sizes for small *δ*_{0} that, when averaged, are dominated by very slow growth in insensitive regions.

FSLE spectra can be interpreted in the context of the results presented in section 3. Static instability manifests as faster growth for larger perturbations. However, the FSLEs show that the rate of error growth is of the same order of magnitude on the different days; a fundamental nonlinearity in error growth of WRF PBL winds is present, regardless of dynamics that might control maximum error levels.

## 5. Summary

Using the WRF model as an experiment platform, we seek an understanding of PBL wind response to small-amplitude uncertainty in land surface properties. Soil moisture is a useful land surface variable in which to introduce uncertainty because the evolution of the atmosphere is sensitive to it. We impose soil moisture perturbations that are consistent with soil moisture spectra analyzes with a land surface data assimilation system, then specify a scale above which the soil moisture is perfectly known and below which uncertainty is saturated. Because we do not have a general description of soil moisture analysis errors, which have scales that vary locally and regionally, generic perturbations smaller than both 16 and 64 km are examined separately. Perturbation magnitudes are smaller than soil moisture errors reported in the literature. In this context, the divergence between model simulations under quadratic metrics is one measure of error growth in response to soil moisture errors. We examine the response during both day and night, and on two days characterized by different static stability and deep-convective activity, to quantify spatial and temporal scales of error growth.

A summary of findings from these experiments follows:

- Atmospheric conditions determine whether details of small-amplitude soil moisture uncertainty are important to the rate of error growth and maximum errors in PBL winds (Fig. 8). Static instability and deep convection, which are sensitive to soil moisture, can act as a mechanism for rapid upscale error transfer in PBL winds. Upscale error transfer can lead to local lack of predictability at scales larger than the scale of deep convection (Figs. 6 –7).
- Small-amplitude soil moisture uncertainty by itself does not produce large errors and upscale error transfer in PBL winds in these cases, whether or not plants are transpiring. Scales of soil moisture uncertainty determine error-growth scales in PBL winds and surface fluxes when the atmosphere is resistant to the upscale transfer of error energy, but when the atmosphere is sensitive the scale of soil moisture uncertainty has little impact after a few hours (Figs. 9 –10).
- Regardless of the presence of deep convection or active transpiration, nonlinear error growth is present at time scales from minutes to hours (Fig. 12). Small-magnitude perturbations grow rapidly in either case, while atmospheric conditions determine nonlinear growth of large-magnitude perturbations.

The results suggest that predictability in PBL winds can be limited by errors in soil moisture specification. This is in itself an intuitive result, but this study shows that the magnitude and spatial scale of error can have little to do with the ensuing PBL wind error under statically unstable atmospheric conditions. When deep convection is present, soil moisture errors at any scale can lead to large errors in PBL wind prediction. On these days substantial improvements to soil moisture accuracy are required to achieve small gains in forecast skill.

Conversely, when the atmosphere is not conditioned for rapid error growth, details of the soil moisture error can determine details of the PBL wind error. Nonlinear error growth is present because the atmosphere can support it, but mechanisms to amplify it and transfer it upscale are absent. These days might be considered resistant to soil moisture errors, at least up to the 64-km errors imposed here. They are also days on which improvements in soil moisture analyses and land surface models may lead to measurable improvements in forecast skill. But the experiments here do not rule out systematic model deficiencies that might lead to systematically poor predictions.

Mesoscale NWP ensemble predictions can be expected to exhibit much greater spread in near-surface winds on days conditioned to support error growth. This agrees with studies showing the sensitivity of deep convection to soil moisture. Deep convection is one conduit for upscale sensitivity in the PBL winds, and simulations on a day with dry soil and no deep convection show little upscale error transfer. One might expect that if normalized, the day-to-day variability in spread of an ensemble constructed by perturbing soil moisture might resemble the day-to-day spread of an ensemble constructed by using different deep-convection or microphysics schemes.

The analysis presented here is far from complete, but it is a reasonable first look at the general response of atmospheric mesoscale error to soil moisture error and offers the basis for further study. We examined a day with convective activity and most of the soil moisture in the domain between the wilting point and field capacity and a day without deep convection and a larger fraction of the soil moisture below the wilting point. Atmospheric and soil dynamics, and the resulting sensitivity to errors in soil moisture, are different between these two days. The frequency of each condition should be quantified to draw conclusions about the potential value of improved soil moisture specifications.

One natural question is whether the WRF might respond similarly to perturbations applied to other land surface characteristics. Soil moisture is somewhat unique because transpiration is a highly nonlinear function of soil moisture. Nonlinearity in the PBL wind response does not appear to depend on whether the vegetation is transpiring, and thus we might expect other parameters to elicit a similar response. As a test, the experiments analyzed above were repeated with perturbations to roughness length *z*_{0} instead of soil moisture. Analysis of 1200 LST perturbations leads to the following conclusions: error growth in surface sensible and latent heat flux and 10-m winds is slower; maximum error levels do not suggest a loss of predictability at any scale, and the total error levels are approximately half of those from soil moisture perturbations; nonlinear error growth is negligible for small-magnitude wind perturbations, but large-magnitude perturbations grow similarly. The response to *z*_{0} is slower, but it does grow, and once wind uncertainty magnitude is great enough, the condition of the atmosphere determines the error growth. Finally, *z*_{0} is static within the short-term simulations and exerts influence by modulating atmospheric forcing on the surface sensible and latent heat fluxes rather than through direct control over the soil state. With further work, one could likely find other dynamic soil variables (such as soil temperature) that yield an atmospheric response similar to that from soil moisture uncertainty.

## Acknowledgments

Comments from A. Barros, E. Patton, and J. Tribbia were instrumental in strengthening the manuscript. I thank W. Yu for providing spun-up land surface states from HRLDAS. This work was supported by the U.S. Defense Threat Reduction Agency–Joint Science and Technology Office for Chemical and Biological Defense.

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Grid specifications and parameterization schemes for the two-domain, one-way nested configuration.

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+ The National Center for Atmospheric Research is sponsored by the National Science Foundation.