Introducing New Metrics for the Atmospheric Pressure Adjustment to Thermal Structures at the Ocean Surface
Abstract
Thermal structures at the sea surface are known to affect the overlying atmospheric dynamics over various spatiotemporal scales, from hourly and subkilometric to annual and O(1,000 km). The relevant mechanisms at play are generally identified by means of correlation coefficients (in space or time) or by linear regression analysis using appropriate couples of variables. For fine spatial scales, where sea surface temperature (SST) gradients get stronger, the advection might disrupt these correlations and, thus, mask the action of such mechanisms, just because of the chosen metrics. For example, at the oceanic submesoscale, around 1–10 km and hourly time scales, the standard metrics used to identify the pressure adjustment mechanism (that involves the Laplacian of sea surface temperature, SST, and the wind divergence) may suffer from this issue, even for weak wind conditions. By exploiting highresolution realistic numerical simulations with ad hoc SST forcing fields, we introduce some new metrics to evaluate the action of the pressure adjustment atmospheric response to the surface oceanic thermal structures. It is found that the most skillful metrics is based on the wind divergence and the SST second spatial derivative evaluated in the across direction of a locally defined background wind field.
Key Points

The standard metrics for the pressure adjustment mechanism is adversely affected by advection

Three new metrics are introduced and tested

The pressure adjustment is detectable in the direction perpendicular to the background wind
Plain Language Summary
The ocean surface is characterized by a range of warm and cold structures that are known to influence the overlying atmospheric flow through different mechanisms. One of these mechanisms involves the variation of sea level pressure that can drive secondary wind circulations according to how the sea surface temperature is distributed in space. To assess whether this mechanism is in action, the colocation of sea temperature maxima (or minima) with zones of wind convergence (divergence) is generally considered. However, the presence of the wind itself has been shown to displace and delay the wind response so that there are cases where the pressure field responds to the sea temperature forcing but this is not detected by the standard metrics. Since pressure variability is generated in all directions, we propose to measure this kind of wind response in the direction perpendicular to the background wind in order to avoid the masking effect of the background wind.
1 Introduction
Sea surface temperature (SST) structures are known to affect the marine atmospheric boundary layer (MABL) dynamics via two main mechanisms: Downward Momentum Mixing (DMM) (Hayes et al., 1989; Wallace et al., 1989) and Pressure Adjustment (PA) (Lindzen & Nigam, 1987). In the DMM physics, spatial variations of SST modulate the atmospheric stability and the vertical mixing of horizontal momentum, resulting in an acceleration (deceleration) of the surface wind over relatively warm (cold) SST patches. In the PA physics, instead, the thermal expansion (contraction) of air over warm (cold) SST patches is responsible for a spatial modulation of the sea level pressure field that, through secondary pressure gradients, drives surface wind convergence (divergence) over warm (cold) SST structures.
The atmospheric response mediated by these two mechanisms has been observed over different time scales and different regions of the world. Notable examples of observations and theoretical modeling of the MABL atmospheric response over annual and multiannual scales include Minobe et al. (2008) and Takatama et al. (2015), both of which focus on a PA interpretation of the atmospheric response over the Gulf Stream. In the same region, and over other western boundary currents, other research has applied a DMM physical interpretation at multiannual (Chelton et al., 2004), seasonal and monthly time scales (Small et al., 2008, and references therein). On the one hand, on scales of the order of few days or even shorter, the works by Chelton et al. (2001), Frenger et al. (2013) and Gaube et al. (2019) have shown that DMM controls the fast atmospheric response over the Tropical Instability Waves of the eastern Pacific cold tongue, over Southern ocean mesoscale eddies and over a submesoscale filament of the Gulf Stream, respectively. Meroni et al. (2020) and Desbiolles et al. (2021), by looking at 25 years of satellite and reanalysis data, have highlighted the prominent role of DMM on daily scales in affecting both the surface wind response and the subsequent cloud and precipitation signature over SST fronts in the Mediterranean Sea. On the other hand, the observational work of Li and Carbone (2012) argues that PA explains convective rainfall excitation over the western Pacific tropical warm pool on daily scales, and the work by Ma et al. (2020) successfully describes the fast atmospheric response to the cold wakes generated by tropical cyclones in terms of secondary circulations controlled by PA (Pasquero et al., 2021). Thus, there is evidence that both mechanisms contribute to the atmospheric response over a large range of spatiotemporal scales.
Most of the idealized model studies, such as Kilpatrick et al. (2014), Skyllingstad et al. (2007), Spall (2007), and Wenegrat and Arthur (2018), show that DMM is more important than PA over small frontal structures and short time scales. However, several authors suggest otherwise. For example, Skyllingstad et al. (2019) demonstrate that PA is the dominating mechanism in the excitation of convective rainfall on daily scales in the tropical ocean, as observed by Li and Carbone (2012). Lambaerts et al. (2013) show that PA is important over hourly time scales, especially in low background wind conditions. Also Foussard et al. (2019) argue that the PAmediated fast atmospheric response has been overlooked in the past because the disruptive effect of the advection on the standard metrics has not been properly considered, as described below.
To measure the action of PA, it is common practice to calculate the correlation coefficient or the slope of the linear fit of the binned scatter plots of SST Laplacian and surface wind (or wind stress) divergence (Lambaerts et al., 2013; Meroni et al., 2020; Takatama & Schneider, 2017). Foussard et al. (2019) highlight the shortcomings of considering these two variables, because advection might shift the atmospheric field with the consequent loss of colocation between the SST forcing and the corresponding MABL response might be lost. To overcome this issue, they propose to use the correlation between air temperature Laplacian, rather than SST Laplacian, and wind divergence, showing that PA is as important as (or even more than) DMM in some environmental conditions. However, air temperature is not easy to observe with satellites and, thus, this approach cannot be followed when analyzing remote sensing data.
It is the objective of this study to define and test three new PA metrics that are robust even in the presence of background wind. In particular, these metrics are based on wind field and SST only, which can be retrieved from satellite measurements and for which there are longterm climate data records (Merchant et al., 2019; Verhoef et al., 2017, e.g.). This work is accomplished using a set of highresolution realistic numerical simulations that have different SST forcing fields. Other than the reference highresolution experiment, there are two runs with enhanced and reduced SST gradients, and a set of runs with different levels of smoothing of the SST field.
Section 2 describes the numerical model and the performed experiments. Section 3 formally introduces the methods and the new metrics. Section 4 describes the results in terms of skills of the metrics, with a focus on the dependence on the strength of the SST gradients and the spatial scales involved. Section 7 discusses and interprets the results and shows examples of application of the new metrics on annual and seasonal statistics derived from reanalysis and satellite data. Conclusions are drawn in Section 8.
2 Numerical Model and Experiments
A set of highresolution realistic simulations with artificially modified SST forcing fields (all constant in time) performed with the Weather Research and Forecasting (WRF) model V3.6.1 (Skamarock et al., 2008) are exploited. Its Advanced Research core that solves the fully compressible nonhydrostatic Euler equations is used. The model exploits an ArakawaC grid in the horizontal and massbased terrain following vertical coordinates. The grid step of the domain of interest is 1.4 km and there are 84 vertical levels. The full setup includes three domains covering the entire Europe at 12 km grid spacing, the Mediterranean region at 4 km, and the Ligurian Sea at 1.4 km. The outer boundary conditions are forced with the ECMWFIFS (European Center for MediumRange Weather ForecastsIntegrated Forecast System) model outputs. The following numerical schemes are used in the simulations: for radiation, the Rapide Radiative Transfer Model (RRTM) for longwaves and the Goddart scheme for shortwaves; for microphysics, the WRF SingleMoment 6Class scheme; for land surface, the fivelayer thermal diffusion scheme; for planetary boundary layer, the MellorYamada Nakanishi Niino level 2.5 scheme, together with the revised MM5 similarity for surface layer. All reference papers to these numerical schemes are available online at https://www2.mmm.ucar.edu/wrf/users/physics/phys_references.html. All simulations are initialized at 0000UTC on the 6th of October 2014 and last for four days. A relatively short experimental time frame keeps computational cost low and enables more experiments to be run. In the present work, only the first output of the simulations, taken at 0100UTC on the 6th of October 2014, is considered in the analysis, for reasons discussed in the next section.
The reference simulation is named CNTRL and is forced with a highresolution SST field, denoted with SST_{0}(x, y), obtained from a realistic eddyresolving ocean simulation integrated with ROMS (Regional Ocean Modeling System) in its CROCO (Coastal and Regional Ocean COmmunity model) version (Debreu et al., 2012; Penven et al., 2006), as described in Meroni, Renault, et al. (2018). The SST_{0}(x, y) field contains smallscale features generated by the ocean dynamics on the numerical grid at 1.4 km. The UNIF experiment is run with a spatially uniform SST field, equal to the spatial mean of the CNTRL SST, indicated as , which is a single number. By taking the difference between the CNTRL and the UNIF SST fields, one obtains the SST anomaly, , which can be increased or reduced to modify the SST gradients. By multiplying the anomaly by a coefficient α and summing back the UNIF SST value, in fact, one gets an SST field with enhanced or reduced SST gradients but with the same mean value as the CNTRL run. The SST fields of the ANML_HALF and ANML_DOUBLE simulations are obtained in this way (with α = 0.5 and α = 2 respectively) to get halved and doubled SST gradients. Note that the gradients are modified just by changing the SST magnitude, and not its spatial scales. The other set of simulations considered, instead, includes simulations with an increasing degree of smoothing of the SST field starting from the CNTRL case, which is not smoothed at all. A Gaussian filter, valid over sea points only, is used to smooth the SST field with a standard bidimensional convolution operation, indicated with ∗. Note that this filter is set to zero after three spatial standard deviations. It is named G_{β} and, correspondingly, the experiments are named SMβ, with β ∈ [1, 2, 4, 8, 16] indicating the standard deviation of the Gaussian filter in km. For small standard deviations, the actual shape of the filter is triangular, as a small number of points are considered. As mentioned above, the SST forcing fields does not evolve in time in any of the simulations. The names of the simulations considered are summarized in Table 1 and for further details the reader is referred to Meroni, Parodi, & Pasquero (2018).
Name  SST forcing field 

CNTRL  
UNIF  
ANML_HALF  
ANML_DOUBLE  
SM1  G_{1} ∗ SST_{0}(x, y) 
SM2  G_{2} ∗ SST_{0}(x, y) 
SM4  G_{4} ∗ SST_{0}(x, y) 
SM8  G_{8} ∗ SST_{0}(x, y) 
SM16  G_{16} ∗ SST_{0}(x, y) 
 Note. Symbols are defined in the main text. SST = sea surface temperature.
3 Methods
By definition, ΔSST_{CNTRL}(x, y) is the SST′(x, y) field introduced in the previous section. In particular, we consider the first hour of the simulations, so that the trajectories of the UNIF run and of the other runs have not diverged too much because of the chaotic nature of the equations and because of different wave propagation features (e.g., in the surface pressure field). Note that, despite having the same boundary conditions, the different simulations develop different turbulent smallscale features, that break the correlation when looking at instantaneous differences. To directly evaluate the PA mechanism in terms of pressure response due to the SST spatial structure we compute the Pearson ρ correlation coefficient between ΔSLP (sea level pressure) and ΔSST from various simulations. When considering ΔSST and ΔSLP, the Pearson ρ is computed using all values from the valid sea points. To test its statistical significance, the number of effective degrees of freedom N_{eff} is obtained using the autocorrelation length of the SST, which, for the CNTRL simulation, is λ_{SST} = 38 km (Meroni, Parodi, & Pasquero, 2018). In particular, , with Ω denoting the area covered by the valid points over the sea. In the null hypothesis of zero correlation, ρ is normally distributed with zero mean and standard deviation equal to (Press et al., 1992). Thus, ρ is statistically significant at the 99% level if it is larger (in absolute value) than .
Longitude φ and latitude θ are defined over a sphere of radius R = 6,371 km.
The strength of using this rotated frame of reference to detect the PA mechanism comes from the fact that pressure is a scalar and produces gradients and, possibly, a dynamical response in all directions. In fact, by looking at the acrosswind direction, it is possible to remove the effects of the largescale advection, which are known to mask the PA signal (Foussard et al., 2019; Lambaerts et al., 2013). Schneider (2020) also exploits a local frame of reference based on the background wind to compute the response and the transfer functions (that can be considered the bidimensional extensions of the coupling coefficient in the physical and the spectral spaces, respectively) to various SST forcings. This proves to be a relevant frame of reference to characterize the atmospheric response.
In what follows, the largescale wind is computed using a bidimensional Gaussian filter on the valid points over the sea with a standard deviation of 10 grid steps (roughly 14 km), unless stated otherwise. A sensitivity to this value is discussed in the next section. A coastal strip of roughly 20 km is removed from the analysis, to avoid including some features that develop in the first few hours of the simulation with numerical waves propagating from the coastlines over the sea. As the next section shows, this does not prevent to detect the local atmospheric response to smallscale SST features.
Two kinds of correlation coefficients are considered: the Pearson ρ and the Spearman r, which is the Pearson correlation coefficient calculated using the ranking of the values, instead of the values themselves (Press et al., 1992). While the Pearson ρ coefficient measures the linearity of the relationship between the two variables under study, the Spearman r measures how much their relationship is monotonic. The statistical significance of the Spearman r coefficient is assessed with a Studentt test (Press et al., 1992). The correlation coefficients are computed either directly on the pointwise values of the relevant fields or on averaged values in classes of percentiles (introduced below). By computing the correlation coefficient using percentile classes one can robustly assess whether the PA mechanism is acting or not, while with the coupling coefficient one can measure the strength of the atmospheric response.
In the literature, binned scatter plots have been used to measure the strength of the airsea coupling, by computing their slope to get the socalled coupling coefficients (Chelton & Xie, 2010; Renault et al., 2019; Small et al., 2008, e.g.,). As the leastsquare estimate of the linear trend is not robust with respect to the presence of outliers, the extreme values in the binned scatter plot can control the value of the coupling coefficient, especially when instantaneous data are considered. To avoid this, the data are organized into percentile bins, so that the statistics are computed over bins with the same number of points, as in Desbiolles et al. (2021). In particular, we compute the mean value and standard error of the dependent variable (y axis) conditioned to the percentile bins of the control variable (x axis). All figures and coefficients shown in this work are computed using 20 bins containing 5% of the points each. The results were tested not to be sensitive to this choice by considering bins with 2% and 10% of the points (not shown).
4 Results
By looking at the correlation between ΔSST and ΔSLP from the CNTRL simulation we can directly evaluate the pressure response to the presence of smallscale SST features because small SLP anomalies are introduced that are highly correlated to the SST anomalies (Figure 2). In particular, a strong correspondence between the ΔSST and ΔSLP fields is visible in the maps of panels (a) and (b). This is confirmed by the high (in absolute value) and statistically significant (>99%) Pearson ρ = −0.94 obtained between the same two fields. This suggests that PA is acting on hourly scales over fine SST structures at midlatitudes, as in the present experiments.
A correlation visually appears also between SST Laplacian Λ of the CNTRL run and Δδ_{CNTRL} fields (not shown), indicating that such smallscale pressure anomalies rapidly force surface wind convergent and divergent cells, in agreement with the physics of PA. However, the correlation between SST Laplacian and wind divergence (without taking the difference with respect to the UNIF simulation, namely, without removing the largescale signal) taken from the same instant of the CNTRL simulation is very low (Figure 3). In particular, from the bidimensional distribution (panel (a)) it is clear how wind divergence is unrelated to the SST Laplacian, especially for very low values of SST Laplacian. The two fields have a very low Pearson ρ, which indicates that the wind divergence variance explained by the linear model as a function of the SST Laplacian is very low (ρ^{2} ∼0.1%). This is physically related to the fact that the atmospheric dynamics is controlled by many processes that have nothing to do with the SST field. A monotonic relationship is not apparent between wind divergence and SST Laplacian (Figure 3b). This is confirmed by the low and nonsignificant (at the 99% level) Spearman r = 0.26 coefficient calculated on the percentile scatter plot. It is worth highlighting that the particular shape of the percentile scatter plot of panel (b) observed here is not a general feature (as it depends on the season, the region and, likely, some environmental conditions) and will not be discussed further.
The bidimensional distribution and percentile scatter plot of the three new metrics are computed (Figure 4). The advantages of considering the acrosswind direction to detect the PA atmospheric response emerge (panels (a) and (b)). In fact, the bidimensional distribution of the acrosswind divergence and the acrosswind SST Laplacian, panel (a), appears to be more symmetric with respect to the origin and shows a slight tilt far from the zero acrosswind SST Laplacian (as highlighted by the linear regression line). The Pearson ρ = 0.038 is still low and not significant at the 99% percent level. The tilt visible in the bidimensional distribution, that corresponds to increasing acrosswind divergence for increasing acrosswind SST Laplacian, becomes more evident in the percentile scatter plot (panel (b)). This is found to have a high Spearman r = 0.85, statistically significant at the 99% level, indicating that the trend is truly positive. It is interesting to highlight that for very negative (positive) acrosswind SST Laplacian, acrosswind surface wind convergence (divergence) is found, in agreement with the action of the physical mechanism.
The use of stretched coordinates does not alter the low correlation values (Pearson or Spearman) for detecting PA (panels (c) and (d)). Moreover, no divergence is ever observed in the percentile scatter plot values, not even at the highest percentiles. This is due to the presence of a largescale negative divergence component, which also emerges in the wind divergence field shown in Figure 3, that causes the mean value to be negative. This is confirmed by the distributions of the wind divergence prime field (panels (e) and (f)). In fact, it appears that the mean wind divergence prime (∼0.25 × 10^{−5} s^{−1}) is an order of magnitude closer to zero than the mean wind divergence (∼−3.5 × 10^{−5} s^{−1}), indicating that the negative bias of the wind divergence and the stretched wind divergence fields is really due to the largescale. Also the us of wind divergence prime (i.e., removing the largescale wind) in the calculation of the correlation coefficients is not enough to highlight the smallscale atmospheric response controlled by PA. In fact, both the Pearson ρ = 0.028 and the Spearman r = 0.43 are relatively low and not significant at the 99% level.
An adjustment to the smallscale wind divergence field caused by PA is noted by looking at the acrosswind SST Laplacian (Figure 5a) and the difference between the acrosswind divergence of the CNTRL case and the UNIF case, Δδ_{s} (Figure 5b).
4.1 Dependence on the Strength of the SST Gradients
The set of experiments that includes ANML_HALF, CNTRL and ANML_DOUBLE are now analyzed. According to the definition of their forcing SST fields, they all have the same spatial mean SST value (equal to the uniform SST used in the UNIF case), with unchanged spatial scales and the SST gradients increasing by a factor of 2. By directly computing the correlation between the ΔSLP and ΔSST fields, we can state that PA is responsible for the atmospheric adjustment irrespective of the strength of the SST gradients, even if they are halved with respect to the CNTRL case. This is proven by the very high (in absolute value) and statistically significant Pearson ρ (at the 99% level) calculated in all three cases (Figure S1 in Supporting Information S1).
The percentile scatter plots of the acrosswind variables, Λ_{s} and δ_{s}, for the ANML_HALF and ANML_DOUBLE runs show that the new metrics based on the acrosswind variables is able to detect a significant correlation (in terms of Spearman r) in both cases (Figure 6). In agreement with the previous results from the CNTRL simulation only, and with the physical understanding of the mechanism, the results from this set of simulations indicate that the surface wind divergence response is enhanced as the spatial variability of SST increases. This implies, then, that the skill of the correlation coefficients to detect the action of the PA mechanism increases with stronger SST variability.
From the bidimensional distributions and the percentile scatter plots of the standard variables, SST Laplacian Λ and wind divergence δ, and of the acrosswind variables, Λ_{s} and δ_{s}, respectively, for the ANML_HALF, CNTRL and ANML_DOUBLE simulations, it appears that the correlations between SST Laplacian and wind divergence are low and nonsignificant, whereas the correlations between acrosswind variables are higher and significant (Figures S2 and S3 in Supporting Information S1. Thus, finescale strong SST variations (on the same spatial scale over which the wind dynamics is resolved) have an imprint in the surface wind divergence field on short time scales. By reducing the masking effect of the advection, in particular by looking at the acrosswind direction, the PA action can be successfully detected, which is not the case if the standard variables (SST Laplacian and wind divergence) are used. Moreover, the fact that the Spearman r increases going from ANML_HALF to ANML_DOUBLE suggests that the presence of stronger SST variability makes this metric more efficient. More on this aspect is developed in the next section.
4.2 Spatial Scale of the Response
The characteristic length scales of the atmospheric response are now considered. In the first place, considering the CNTRL simulation, two things can be tested: (a) the skills of the standard metrics (based on Λ and δ) as a function of the standard deviation σ of a Gaussian filter used to smooth the SST Laplacian and wind divergence fields themselves, and (b) the skills of the acrosswind metrics (based on Λ_{s} and δ_{s}) as a function of the standard deviation σ used to define the largescale background wind field.
The Spearman r between the smoothed standard variables shows that for a very local smoothing (small σ), the correlation is relatively low, ∼0.4, while a peak in the correlation is reached with σ between 25 and 30 km (Figure 7). This is interpreted to be due to a reduced masking effect of the advection when the fields are smoother, linked to a better match between the length scale of the PA atmospheric response and the length scale of the SST forcing, as discussed more in detail in the next section. In the same panel, the acrosswind variables show a high and significant correlation up to σ ∼25 km. With σ between 25 and 40 km the correlation drops and after 40 km it is no longer significant at the 99% level. In the limit of very large σ, the correlation is expected to be similar to the value of the nonfiltered standard metric (correlation between the SST Laplacian and the wind divergence), as a uniform background wind is used to compute the acrosswind derivatives and no information on the local structure of the flow is retained. This indicates that the metrics based on the acrosswind variables is able to detect the PA signal for σ < 25 km.
By considering the set of simulations with a smoothed SST field, the SMβ set of experiments, the skills of the new metrics when the SST gradients get weaker both because the SST variability decreases and because their spatial scales increase can be tested. Note that the standard deviation of the filter applied to the SST forcing β is completely independent from the standard deviation of the filters applied to the diagnostic fields σ. We verify that the direct atmospheric response in terms of pressure, measured by the Pearson ρ correlation between ΔSLP and ΔSST is strong and significant in all SMβ cases. It is found that the correlation is always lower than −0.91. Thus, despite the SST first and second derivatives get weaker because of the spatial smoothing, the presence of a nonuniform SST introduces a direct atmospheric response in terms of surface pressure. The maps of ΔSLP and ΔSST also confirm the strong correspondence of the two fields (not shown).
Considering smoother SST forcing fields, a consistent behavior of the smoothed standard variables emerges (Figures 7b–7f). In fact, it is always found that a smoothing with a σ of 20–30 km is needed to reduce the advection effect and get the peak in correlation suggesting that the SST forcing at these scales is detected by the atmospheric dynamical response. In terms of acrosswind variables, instead, it emerges that when the forcing SST field does not have any smallscale feature (starting from SM4, panel (d), and for higher β), the wind field is not constrained by the SST and the correlation is not significant for σ < 20 − 30 km. For higher σ, instead, the Spearman r of the acrosswind variable tends to the Spearman r of the nonsmoothed standard variables (SST Laplacian and wind divergence), as previously discussed. This confirms that the metrics based on the acrosswind variables does not detect any smallscale atmospheric response in the case where no smallscale SST forcing is present, which is important to show for the definition of a new metrics.
Note that in the definition of the acrosswind divergence δ_{s} and the primed wind divergence δ′ the background wind field is removed. Thus, considering these variables instead of the wind divergence δ is a form of highpass filter whose cutoff length is determined by σ itself. The larger the σ, the smoother the background wind field, but the wind divergence fields always have a smallscale component. So far, the background wind Gaussian filter has been defined with a standard deviation σ of 10 grid points (equivalent to 14 km), but its values can be used to select the scales of the atmospheric response of interest in the δ_{s} and δ′ fields.
The Spearman r correlation between the percentile scatter plots of the stretched SST Laplacian and the stretched wind divergence has a very weak dependence on the σ used to determine the background wind field for both the CNTRL and all the SMβ runs (not shown). This happens because in the calculation of the stretched variables the largescale wind is not removed and there is no highpass filter behavior. For all cases, then, the correlation is never significant at the 99% level. Instead, we do not show the Spearman r correlation between the SST Laplacian and the wind divergence prime δ′, because its behavior as a function of σ is similar to the acrosswind variables one, with generally lower correlation values.
5 Discussion
The new metrics based on the acrosswind variables has been shown to detect the PA signal over hourly time scales in the midlatitudes (Figure 4). This is in agreement with the results of Lambaerts et al. (2013). In their work, they are able to show it by computing the standard metrics (correlation coefficient between the vertical wind velocity, closely related to the horizontal wind divergence, and the SST Laplacian) in some idealized numerical simulation with absent or very weak (1 m s^{−1}) background wind. The fact that here the correlation between the standard variables is low can be explained by the presence of a nonzero background wind (whose histogram is shown in Figure 8). It ranges from 0 to 5 m s^{−1}, with a mean value of 3 m s^{−1} over the sea in the instant considered. In agreement with the arguments presented by Foussard et al. (2019), the presence of a nonzero mean wind breaks the correlation between SST Laplacian and wind divergence.
By considering the simulations with smooth SST fields, then, it has also been shown that when the small scale SST forcing is not present, the new acrosswind metric does not detect any atmospheric response, as expected. In fact, as the spatial scale of the SST structures increases (corresponding to high β in the SMβ simulations), the scales of the SSTinduced pressure gradients also increase. This means that, at fine scales, the SST structure does not produce any pressure gradient that can alter the wind field, and, thus, the finescale wind variability cannot be constrained by the SST. This has been tested by changing the standard deviation of the Gaussian filter used to calculate the background wind speed σ and considering all simulations of the SMβ set (Figure 7).
In the literature, the characteristic time scale of the PA mechanism is written as h^{2}/K_{T}, where h is the MABL height and K_{T} is the thermal eddy turbulent coefficient (Small et al., 2008). Physically, this corresponds to the time required for a nonnegligible pressure anomaly to develop, which is controlled by the temperature mixing in the MABL. By looking at the CNTRL simulation, the MABL height is between 300 and 1,400 m, whereas a typical midlatitude value for K_{T} is 15 m^{2} s^{−1} (Redelsperger et al., 2019). By multiplying the PA time scale by the typical wind speed U_{0}, one gets the length scale over which PA produces a wind response (Small et al., 2008). In particular, using the mean wind speed of U_{0} ∼3 m s^{−1} of the instant of the simulation considered (see Figure 8), the PA length scale L_{p} ∼ U_{0}h^{2}/K_{T} is in the range between 15 and 360 km. The σ of the filter that maximizes the Spearman r between the smoothed SST Laplacian and the smoothed wind divergence, which is around 30 km, falls in this range. In particular, as the extent of the Gaussian filter is actually 3 times its standard deviation, we can consider that the length scale of the structures that maximizes the SST Laplacian and wind divergence correlation is roughly 100 km, which is very close to the mean value of L_{p} ∼120 km. This suggests that the masking effect of the advection on the correlation between SST Laplacian and wind divergence is reduced when some smoothing is performed on the wind field and when the scales of the forcing SST are of the same order as the PA length scale.
In other words, the PAmediated secondary circulation develops in response to the underlying SST structures on a length scale L_{p}, which, in the direction of the wind, is large compared to the typical SST structures. Thus, as the response of the air moving with the flow is integrated over the small scale SST variability, it is only sensitive to the smoother and larger scale thermal features. In the acrosswind direction, the advection U_{0} tends to zero and, thus, the PA length scale L_{p} tends to zero as well. For this reason, the spatial response mediated by PA can be detected over very small scales by the newly introduced metric, as previously demonstrated.
None of the two other metrics is found to be skillful. In fact, the use of the coordinate stretching does not correspond to any increase in the correlations, because there is no selection of the small scales (accomplished in the other cases with the subtraction of the background wind). Removal of the largescale wind before computing the wind divergence result in a modest improvement with respect to the full wind divergence field. This is explained by the presence of the effects of the largescale advection, which keeps the skills of this metrics lower than the acrosswind one. This corresponds to the fact that the integral PAmediated atmospheric response is realized over relatively large L_{p} scales.
The new metric based on the acrosswind variables can also be applied to some highresolution satellite data. The daily L4 ESA CCI (European Space Agency Climate Change Initiative) SST analysis product v2.1 (Good et al., 2019; Merchant et al., 2019) and the instantaneous L2 coastal METOPA ASCAT (METeorological OPerational satelliteA Advanced SCATterometer) wind field CDR (Climate Data Record) product (Verhoef et al., 2017) are considered. The ESA CCI SST analysis is given on a regular 0.05° grid and the METOPA ASCAT wind on its irregular alongtrack grid at 12.5 km nominal resolution.
Considering all the wind swaths within the spring season (from the 1st of March to the 31st of May 2010) over the Mediterranean Sea, the seasonal percentile scatter plots for the standard metrics (SST Laplacian and wind divergence) and the acrosswind variables can be computed (Figure 9). It appears that a different response is detected according to the variables considered. In particular, no relationship between the wind divergence and the SST Laplacian is detected, in agreement with previous studies such as Meroni et al. (2020) and Desbiolles et al. (2021). However, a significant Spearman r correlation is found between the acrosswind variables, suggesting that PA is actually at play, as found from the numerical simulations presented in this work. Thus, concluding that the PA mechanism does not control the atmospheric wind response over the Mediterranean Sea might be incorrect just because the signal is masked by advection, as discussed in the previous sections. A full characterization of the wind response using these data goes beyond the scope of the present work and will be considered in a future work. Here, we can state that the newly defined acrosswind metric is able to detect a PAmediated signal even in high resolution remote sensing observational products.
Finally, we can verify a posteriori that the improved detection skills of the acrosswind metrics with respect to the standard one emerge in all wind conditions and irrespective of the region considered. This is accomplished by analyzing 1 year (2007) of global daily ERA5 reanalysis data (Hersbach et al., 2020). In particular, the standard and the acrosswind metrics are computed for different classes of background wind (between 0 and 5 m s^{−1}, between 5 and 10 m s^{−1}, between 10 and 15 m s^{−1}, and above 15 m s^{−1}). It appears that the signal in the acrosswind metrics emerges for all background wind conditions over the globe (Figure 10), which enables to generalize the results of the present simulations, that are limited to relatively weak wind conditions and over a small region.
6 Conclusions
The PA mechanism is mostly known in the literature to produce a wind divergence response over large SST structures and relatively long scales, namely seasonal and annual (Minobe et al., 2008; Takatama et al., 2015). Evidence of its control on the wind divergence over finescale SST structures and short time scales has been detected either in very low or absent background wind environments (Lambaerts et al., 2013), or exploiting correlation coefficients between wind divergence and air temperature (Foussard et al., 2019), which is not easy to observe from satellites. Advection has been proposed to be the main responsible for the breaking of the correlation between SST Laplacian and wind divergence (Foussard et al., 2019), which is one of the standard PA metrics (Minobe et al., 2008; Small et al., 2008).
In this work, we introduce and test three new metrics to detect the fast action of PA exploiting SST and wind field data, only. The skills of the new metrics are evaluated using a set of highresolution realistic numerical atmospheric simulations with appropriately modified SST forcing fields. In particular, the presence of a simulation with a uniform SST field enables to directly look at the effects of the SST spatial structures on the MABL dynamics. Among the proposed metrics, only the one based on the correlation between the acrosswind SST Laplacian and the acrosswind divergence, so that the masking effect of the largescale wind advection is reduced, is able to detect the PAmediated atmospheric response. This approach exploits the fact that pressure is a scalar and it can produce gradients in all directions. A significant Spearman r correlation between the acrosswind SST Laplacian and the acrosswind divergence is found when the SST forcing field has smallscale spatial structures, whereas no correlation is detected when the forcing SST field is smoothed. This is in line with the physical interpretation of the characteristic length scale of the PAmediated response, L_{p} ∼ U_{0}h^{2}/K_{T}, which is large in the alongwind direction, L_{p} ∼100 km in the present setup, and tends to zero in the direction perpendicular to the background wind, where U_{0} tends to zero. This explains why the new metrics is able to detect the PAmediated response over short spatial scales. If the focus is on larger spatial scales, of the order of the PA adjustment scale L_{p} ∼100 km, also smoothing the SST Laplacian and the wind divergence fields can recover the correlation. This extends the findings of Lambaerts et al. (2013) to higher background wind conditions and confirms the results of Foussard et al. (2019). Global daily ERA5 reanalysis data also show a posteriori that the acrosswind metrics has improved skills in detecting the PAmediated response with respect to the standard metric, irrespective of the background wind conditions.
An example of application of these new metrics to highresolution satellite data in the Mediterranean Sea shows that by looking at the acrosswind direction, a PAmediated wind response emerges on subdaily time scales, which has never been observed before using the standard metrics (Desbiolles et al., 2021; Meroni et al., 2020). Future efforts devoted to characterize the spatiotemporal variability of the PAmediated response using satellite data at high resolution from current and future missions (such as those proposed in the European Space Agency, ESA, Earth Explorer X Harmony, ESA (2020)) will allow to better characterize airsea feedbacks and to properly parameterize them in climate models.
Acknowledgments
The authors acknowledge support from the project JPI Climate Oceans EUREC4AOA, Progetto Dipartimenti di Eccellenza, funded by MIUR 20182022 and the European Space Agency (ESA) contracts 4000135827/21/NL/FF/an and 4000134959/21/NL/FF/an. A. N. M. is supported by ESA as part of the Climate Change Initiative (CCI) fellowship (ESA ESRIN/Contract No. 4000133281/20/I/NB). F. D. is supported by ESA contract n. 4000127657/19/NL/FF/gp and by HPCTRES Grant No. 202010. The authors thank David E. Atkinson and an anonymous reviewer for the fruitful comments during the revision. Open Access Funding provided by Universita degli Studi di MilanoBicocca within the CRUICARE Agreement.
Open Research
Data Availability Statement
The WRF model outputs of interest can be downloaded from https://doi.org/10.5281/zenodo.5534305 (Meroni, 2021a). The L4 ESA CCI SST analysis product v2.1 is available from the Centre for Environmental Data Analysis (CEDA) archive (Good et al., 2019). The L2 coastal ASCAT METOPA CDR wind field product (Verhoef et al., 2017) is available from the NASA JPL PODAAC platform (EUMETSAT/OSI SAF, 2018). ERA5 data have been downloaded from the Copernicus Climate Data Store (Hersbach et al., 2018). The analyses of the WRF model outputs have been carried out with a Jupyter Notebook available at https://github.com/agonmer/meroni_etal_JGRA_2022.git (Meroni, 2021b).