1. Introduction
The ocean surface exerts strong control on the atmospheric boundary layer through momentum, heat, and moisture exchange across the ocean–atmosphere interface. Realistic ocean modeling places a premium on this influence, but common practices can omit it. For example, specifying fluxes at the ocean surface misses this connection; thereby, for example, locating strong oceanic heat loss away from the subsurface structure with which they should be associated. Even high-resolution, high-accuracy scatterometer wind stress data can be uncorrelated with the surface velocity expression of a free-running ocean simulation. Another common approach consists of specifying atmospheric temperature, humidity, and wind and then diagnosing from them air–sea fluxes with a bulk equation. However, the ocean naturally develops scales much finer than the resolutions currently available from all atmospheric reanalysis products, and the imprint of these scales on the exchanges, with any subsequent feedbacks on the atmospheric variables, are lost. Given the low heat capacity of the atmosphere, the reaction of the atmosphere can be strong, thus influencing later heat exchanges and precipitation.
To fully capture the ocean–atmosphere connection at the interface would require a fully coupled ocean–atmosphere model. It is an open question if such a thing currently exists; however, even if it did, the computational burden associated with its use would be restrictive. It is thus of practical value to have alternatives that replicate at least some of the important coupling features.
Seager et al. (1995, hereafter SBK) in an insightful paper, recommended a partial solution to this problem. They proposed the use of a thermodynamically active, but dynamically passive, atmospheric boundary layer. The wind was specified, relieving the need to compute atmospheric dynamics, and with an assumption of rapid equilibration of the atmospheric temperature and humidity, the atmospheric state was diagnosed. Applications in models showed clear improvement in the flux structure relative to other products. Such an approach provides a means of overcoming many of the leading order omissions in ocean modeling associated with either flux specification, the use of bulk formulas, or relaxation boundary conditions, while simultaneously retaining the computational efficiency and flexibility inherent in ocean-only modeling.
Our goal here is to modernize the SBK thermodynamic atmospheric boundary layer model. The four primary distinctions between our approach and that of SBK are as follows: 1) the use of modern flux algorithms, 2) abandonment of the equilibrium assumption, 3) calculation of an accurate freshwater flux, and 4) ease of migration to parallel computing. Specifically, the modular model design permits the user to either develop a subroutine using the flux algorithms of their choice, or to choose from the methods of Large and Pond (1982, hereafter LP82) or the Coupled Ocean–Atmosphere Response Experiment, version 3 (COARE3; Fairall et al. 2003). We discard the equilibrium assumption because modern computing involves specifying atmospheric data typically at rates of a few times per day, as opposed to the use of lower-frequency climatological data as was the custom at the time of SBK’s original paper. Coupled with this is the prediction, rather than diagnosis, of the atmospheric state, which removes the need to solve an elliptic equation. The latter, aside from suppressing “weather” responses, are difficult to efficiently migrate to parallel platforms. The impact of the equilibrium hypothesis has been studied by Hazeleger et al. (2001). They added a parameterization of atmospheric storms to SBK and found significant impacts in several regions of the Pacific. The use of daily wind (as opposed to a monthly climatology) greatly modifies latent and sensible heat fluxes as well. Indeed the sensible heat fluxes scales as ∝uΔT, with u as the magnitude of the surface wind and ΔT as the temperature difference between the ocean and the atmosphere. When suppressing the variability of u and ΔT by taking a mean value, it appears that the fluxes are systematically underestimated, especially when the variability of u is large (not shown).
Our model, named Cheap Atmospheric Mixed Layer (CheapAML), has been implemented in the Massachusetts Institute of Technology general circulation model (MITgcm) and is available with the standard distribution as a package. This FORTRAN code computes the forced atmospheric temperature and relative humidity tendency terms; it can be downloaded and modified for any model.
In principle, this atmospheric module can be applied to any model configuration (e.g., coarse grids, eddy resolutions, and for regional to global configurations) without tuning. However, we show in the following that in certain regions, this model can introduce a drift due to unmodeled physical processes. In such regions (mainly the tropics), it is preferable to adjust some parameters or to introduce a relaxation toward climatological values. Another limitation might be the availability of a fine-resolution wind field (spatial and temporal) for meso- or submeso-scale ocean applications. We discuss the effect of wind–SST interaction (Small et al. 2008) even though this has not been implemented in CheapAML.
The paper is organized as follows. The model is described in section 2. In sections 3 and 4 we describe the results of two verification experiments. First a regional experiment in the Gulf Stream is presented, followed by a global modeling application. The conclusions are given in section 5. The appendix compares several methods currently available to compute latent and sensible heat flux.
2. Model equations
a. Main equations
We also employ the simplifying assumption that the atmospheric boundary layer is described by a known, but possibly variable, thickness h. The model also employs a specification of the tracers s over land and allows time dependence in those specifications.
Precipitation is generally one of the most difficult atmospheric variables to predict and a model of this simplicity will suffer when applied to the wide variety of realistic precipitative conditions. Precipitation in the boundary layer only enters the water vapor budget as a small correction and we chose to not retain it. We only compute it as a diagnostic field for the ocean freshwater budget.
Here we describe a parameterization that is physically justified and that has performed reasonably well in tests. The applications in the next sections illustrate its strengths and weaknesses, and we provide methods by which the weaknesses can be addressed.
b. Air–sea turbulent fluxes
c. Boundary values and relaxation
CheapAML also requires the specification of the tracer s on the lateral boundaries (when implemented in an open boundary configuration) and on land, and allows time dependence in those specifications. The current version of CheapAML does not include a land module. Instead, the temperature and humidity are strongly relaxed toward provided, and possibly time-varying, fields. This is the primary role of the last term of (1); thus, default specifications are 1/λ = 2 h over land and λ = 0 over the ocean. A secondary use of the λ parameter is to nudge the model toward observations in the manner of data assimilation and thus correct for missing model physics. The quantity λ can be specified as a field variable to facilitate this, with the limit of large λ everywhere converging to the classical case where atmospheric variables are specified and fluxes are computed using bulk formulas.
d. Height of the boundary layer
The vertical fluxes of temperature and humidity given in (4) and (9) both depend on the height of the boundary layer h. The default configuration of CheapAML assumes a uniform h value of 1000 m, a value that provides demonstrably useful fluxes. However, our experience is that it is often advantageous to provide CheapAML with readily available boundary layer thickness data. For example, for global configurations, a constant, single value for the boundary layer height fails to return broadly accurate fluxes and marine boundary layer behavior. In the extratropics, h varies seasonally from lower than 500 m in the summer to more than 1200 m in the winter. In the tropics h remains between 600 and 1000 m all year long. CheapAML is designed to optionally accept temporally varying h values, such as are available from the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) dataset.
In all the following experiments, we use the daily varying climatology of the boundary layer height provided by the ERA-40 dataset.
3. A regional experiment
a. Mean and variability
To illustrate and assess CheapAML performance, we apply our model to the separated Gulf Stream (GS), where ocean eddying, weather, and flux are particularly strong. Here, the SST can vary by more than 10°C over 100 km, which is precisely the characteristic grid space of common reanalysis datasets [Kalnay et al. (1996) for the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis and Uppala et al. (2005) for the ERA-40].
We examine here how CheapAML compares to ERA-40 atmospheric temperature, humidity, and fluxes in a confined regional configuration with prescribed SST. The boundaries of the chosen domain are 75°–45°W in longitude and 34°–45°N in latitude. “Truth” fluxes have been computed by specifying ERA-40 wind velocity, temperature, and humidity at 6-hourly intervals, with linear interpolation to times in between. CheapAML fluxes are computed by specifying boundary atmospheric variables and default λ values. The underlying SST also comes from the ERA-40 dataset. Here again, at each time step, the SST is interpolated between the two nearest records. All fields are spatially interpolated to the finer resolution of 
Figures 1 and 2 summarize the evolution of the atmospheric variables for January 2007. We plot the monthly mean CheapAML and ERA-40 atmospheric temperature and humidity. The mean temperature difference between CheapAML and ERA-40 does not exceed 0.7°C (Fig. 1). There is a cold bias of CheapAML relative to the truth over the GS path and a warm bias elsewhere. There is also good agreement in the variability pattern of temperature (computed on a daily basis) in both cases. It is slightly overestimated by 0.7°C over the cold side of the GS and underestimated by 0.5°C over the warm side.
(top left) Mean lower-layer atmospheric temperature (°C) and (top right) standard deviation of daily values from that mean in January 2007 (daily data from ERA-40). (middle) As in (top), but reconstructed with a one-month CheapAML integration (starting date: 1 Jan 2007). (bottom) Difference (°C) between the (middle) and (top).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
As in Fig. 1, but for humidity (g kg−1).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
We draw similar conclusions when looking at the mean and standard deviation of daily values of humidity (Fig. 2). The mean humidity is overestimated outside of the GS path by 0.8 g kg−1 in accordance with the temperature bias. The pattern of variability has a similar shape in ERA-40 and CheapAML. In the latter, it is underestimated everywhere with a maximum over the warm side of the front.
The heat and moisture fluxes (not shown) also show similar results; that is, differences are in the 10%–20% range, which is well within the uncertainty of bulk flux parameterizations (see also the appendix). Comparable results are found if fluxes are computed using the SBK approach. An improvement of CheapAML relative to SBK lies in the variability pattern. The variability of temperature and humidity are underestimated over the entire region by more than 1°C and 1 g kg−1, respectively by SBK. This relatively weak variability is consistent with the equilibrium hypothesis, and reflects a lack of weather on air–sea exchange.
b. Representation of the extreme events
To further examine the time variability, we compare several time series of temperature taken in the middle of the domain (39°N, 62°E). In Fig. 3a, we compare a 3-month time series of temperature from the ERA-40 (thick black line), CheapAML (blue line), and SBK (red line). Both SBK and CheapAML are in good agreement with ERA-40, with correlations of 0.98 and 0.99, respectively. However, bias appears especially in the representation of the extremes (cold and warm events). We plot in Fig. 3b the probability density function (PDF) of these three time series using the same plot convention. CheapAML represents correctly both warm and cold events, if slightly overestimating the cold. The global shape of the ERA-40 PDF is well reproduced—especially the bimodality that is observed during this period. The PDF of the SBK simulation is much more peaked at the center of the distribution. These results reflect the SBK equilibrium hypothesis.
(a) Time series of the atmospheric temperature in the middle of the regional model (cf. Figs. 1–2) from 1 Jan to 31 Mar 2007. The thick black line is the ERA-40 temperature, blue is CheapAML, and red is SBK. (b) PDF of these three time series. (c) PDF of the humidity time series using the same color conventions.
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
The PDF of humidity at the same location and for the same period is plotted in Fig. 3c. Whereas the SBK simulation has a peaked distribution of humidity, CheapAML recreates more accurately the observed distribution.
The sensible and latent heat flux time series at this location are also well captured by both CheapAML and SBK (correlation above 0.98 in each case). We only show in Fig. 4 the PDF of these time series. The thick black line corresponds to heat fluxes computed using surface fields from ERA-40 but applying the COARE3 formulation. The dashed line corresponds to raw heat fluxes extracted from the ERA-40 dataset. The comparison illustrates the differences inherent to state-of-the-art flux algorithms, which are considerable (see also the appendix). Once again CheapAML yields a better PDF when compared to the COARE3 implementation than does SBK.
PDF of the (a) sensible and (b) latent heat fluxes for the first three month of 2007. Thick black line corresponds to the fluxes computed using the ERA-40 fields with the COARE3 algorithm, the dashed line is the raw ERA-40 fluxes, the blue line is the fluxes using CheapAML, and the red line represents SBK.
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
c. Impact on the oceanic circulation
We now extend the simulations to a more realistic ocean experiment. Our intent is to illustrate the impact of the two atmospheric mixed layer (AML) formulations (SBK and CheapAML) on the oceanic structure. We still focus on the region of the separated GS and use the MITgcm (Marshall et al. 1997) with open boundary conditions. The model obtains boundary data from the Hybrid Coordinate Ocean Model (HYCOM) ocean reanalysis dataset (Chassignet et al. 2007). The starting date is 1 January 2007 and the model is run for 1 month. The spatial resolution is 
We first compute the air–sea exchange using SBK and then using CheapAML. Since the oceanic states quickly differ after several days, it is not useful to compare the mixed layer depth at a specific location. We plot in Fig. 5 the evolution of the mixed layer depth averaged over the entire domain. The SBK integration is plotted with a dashed line, the CheapAML integration is plotted with a thin line, and the HYCOM reanalysis is plotted with a thick line. During this month the mixed layer depth increases everywhere in both simulations, as expected. The first 15 days are well reproduced in the CheapAML run whereas the last 15 days are better reproduced in the SBK run. After one month, we observe a notable difference of 50 m in the two runs whereas the HYCOM reanalysis lies in between these two estimates. The only possible reason to explain these two evolutions lies in the differences in the heat and freshwater fluxes. This illustrates the discrepancies of the oceanic state that occur due to uncertainties in forcings.
(a) 30-day time series of the mean mixed-layer depth for three oceanic states: HYCOM reanalysis (thick line), CheapAML run (thin line), and SBK run (dashed line). (b) Corresponding net heat flux averaged over the domain. The additional curve (dashed–dotted) corresponds to the OAFlux value.
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
The spatial average of net heat flux is given in Fig. 5. The associated mean values for this period are: 303 W m−2 for CheapAML, 486 W m−2 for SBK, 357 W m−2 for HYCOM, and 304 W m−2 for objectively analyzed air–sea fluxes (OAFlux; Yu and Weller 2007). The favorable comparison between OAFlux and CheapAML partly reflects their common basis in the COARE3 algorithm, but they do use scatterometer winds while we use ERA-40 winds. The small difference between our numbers is consistent with the idea that local feedbacks on fluxes due to wind modifications by the oceanic mesoscale are not a major systematic error.
The overall character of the means is consistent with the mixed layers diagnosed in the three cases. The interpretation of the differences between The CheapAML and SBK with HYCOM is unclear since the latter is an assimilative product.
Nevertheless, we can associate the rapid deepening of the mixed layer with the peaks of the net heat flux. SBK, which consistently predicts a higher net heat flux, has a more pronounced mixed layer deepening rate. While all the net heat flux curves are well correlated (above 0.9 for all pairs), the magnitude of the storm peaks are very different. For the first day of the run (same ocean state for all experiments), we note a difference of 300 W m−2 between OAFlux and HYCOM. The maxima in net flux during storms are either reached by HYCOM or SBK. In contrast, HYCOM, OAFlux, and CheapAML agree well at the minima (7 and 16 January)—with a difference of less than 50 W m−2 between them.
4. Global experiment
In the previous section, we demonstrated the benefits of using CheapAML in a regional model. We here show how CheapAML can be used for global experiments, where we again use prescribed SSTs. In a configuration corresponding to the ERA-40 setup (1.125° resolution), we constrain the SST to follow the ERA-40 SST field (perfect ocean model experiment). Over the ocean, we let the air temperature and humidity adjust via CheapAML. The model is initiated in January 2000 and run for 13 months; we focus on the last month. We do not have a varying land–sea mask and treat sea ice points as land points. Atmospheric forcing is drawn from ERA-40. CheapAML is deployed using standard λ values and ERA-40 boundary layer heights.
a. Atmospheric variables
We compare the mean temperature computed in January 2001 with that in ERA-40 (Fig. 6). The bottom panel of Fig. 6 corresponds to the differences between the middle and the top panels. The largest bias is observed in the Northern Hemisphere. More generally, when looking at the June–July–August maps, we conclude that the largest bias occurs in the winter hemisphere. We observe a cold bias near the western boundaries whereas the center of the Atlantic and Pacific are subject to a warm bias. In the tropics, we observe a warm bias in the region of strong convection. These errors reflect processes not modeled here but do not exceed 1.5°C.
(top) Mean atmospheric lower-layer temperature (°C) in January 2001 (data from ERA). (middle) Reconstructed temperature (°C) in January 2001 with CheapAML starting 1 Jan 2000. (bottom) Difference (°C) between the (middle) and (top).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
Figure 7 focuses on the mean humidity field. Again, we note that the patterns in the two top panels are in accordance. The humidity maximum in the tropics is well reproduced as are the specific patterns in the extratropics. The bottom panel is the difference between the simulated and observed humidity. The maximum differences reach ±1 g kg−1 [typically O(5%–10%)] in the tropics as well as in the extratropics. The humidity is mainly overestimated in the northern and southern part of the oceanic basins whereas it is mostly underestimated in the tropics.
As in Fig. 6, but for humidity (g kg−1).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
The regions where the air temperature or humidity fields are not well estimated reveals the zones where the outgoing flux at the top of the boundary layer or the precipitation is not properly described in our model. Other physical phenomena are thus at work in these regions (convection, clouds, and vertical motion). Knowing these discrepancies, one could structure λ so as to minimize model bias.
The precipitation field for January 2001 is plotted in Fig. 8. Although the parameterization mentioned in section 2 is extremely simple, some skill is observed. The two top panels of Fig. 8 argue that the global patterns of the convective precipitation are well reproduced in the tropics. However, the large-scale precipitation in the northern Pacific and Atlantic are underestimated. These precipitations are associated with the position and strength of the storm track and cannot be easily reproduced using our single layer model.
As in Fig. 6, but for precipitation (mm day−1).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
b. Net heat flux
Figure 9 compares the net heat flux computed by CheapAML with that given by OAFlux (Yu and Weller 2007) for January 2001. CheapAML captures correctly the large-scale pattern (enhanced heat flux over the western boundary current in January and north–south asymmetry), but appears to underestimate the heat flux. In the Northern Hemisphere where the mean heat flux is positive, it is underestimated by 20–70 W m−2. This pattern is clearly related to the temperature pattern anomaly observed in Fig. 6: an atmosphere that is too warm prevents strong air sea flux in that region. The difference seen in the Southern Hemisphere has a pattern that is very similar to the humidity differences: a dry mixed layer leads to a larger evaporation and thus an increase of the net heat flux. In July the situation is reversed (not shown). This is also in accordance with the temperature and humidity differences observed in July (not shown): the Northern Hemisphere Qnet is large enough. Note also that some differences might also be explained by the longwave parameterization [see (8)], although this heat source remains small compared to the sensible and latent heat flux.
As in Fig. 6, but for Qnet (W m−2).
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
5. Conclusions
a. Summary
We introduce here a simple atmospheric boundary layer model for the computation of air–sea exchange in ocean-only modeling. In the boundary layer, temperature and humidity are advected by a prescribed wind. Temperature and humidity adjust with the underlying SST mainly through sensible heat and evaporation. The value of this model is to capture part of the nonlocal feedback of the ocean surface on air–sea exchanges, while stopping well short of computing a full coupled ocean–atmosphere model. We believe that for an oceanic model, it is preferable to use CheapAML than to prescribe the temperature and humidity (or fluxes) from a reanalysis dataset: as soon as the oceanic state deviates from the observed state, the reanalysis temperature and humidity fields and the oceanic state are not related anymore. The computational cost of using CheapAML is minimal, and does not materially increase the execution time of the model run. Furthermore, CheapAML captures the “weather” impacts of the atmosphere on air–sea exchange with improved fidelity relative to its predecessor, SBK.
Using a regional and a global configuration we tested the skills of CheapAML. In a small region subject to large spatial variations of SST, we show that this slab atmospheric model is able to accurately reconstruct the mean temperature and humidity fields as well as their variability. Analyzing several time series of atmospheric tracers and fluxes at given locations, we argue that CheapAML reproduces the mean as well as the extreme events correctly. The extreme events (e.g., for cold air outbreak) are of great importance for the oceanic dynamics. We illustrate this impact using the evolution of the mixed layer depth when the ocean is subject to these different fluxes. When deployed globally, zones appear where temperature and humidity are subject to biases. These biases, although small, are inherent to the simplifications performed to construct this model, and can be reduced through nudging.
The main differences between this model and its predecessor SBK are the elimination of the equilibrium assumption and the provision of a water budget. Here, we explicitly integrate in time the equation of evolution of temperature and humidity. We also updated the computation of the air–sea fluxes using a more recent formulation (Fairall et al. 2003). An accurate computation of the air–sea fluxes is in fact the primary goal of this study. The freshwater flux budget exhibits similarity with the observed freshwater budget although a better representation of precipitation might help to increase the accuracy of this forcing. Moreover, we propose here a fully parallel code whereas the computation of SBK requires the knowledge of atmospheric variables in the entire domain and is thus harder to parallelize.
b. Remaining issues
Among issues for future development are the development of atmosphere–land and atmosphere–sea ice modules. Such regions are handled by means of strong relaxation toward specified values; these are clear areas for improvement.
Clouds are also not parameterized in this model. It is, however, possible to adjust the solar input to mimic the presence of clouds, although not dynamically.
Several studies [see Small et al. (2008) for a review] indicate that there is a correlation observed between the wind speed and the SST; the wind being accelerated over warm SST. This interaction could also lead to some possible refinement of our model, especially in regions of strong SST fronts or eddies. Pezzi et al. (2004) and Jin et al. (2009) proposed parameterizations of the wind–mesoscale eddies interaction. According to their results, the detailed structure of the oceanic eddies is affected by this interaction. How this may impact the large-scale ocean circulation remains to be seen.
c. Practical use of CheapAML
We recommend the use of CheapAML via the MITgcm (Marshall et al. 1997), where it was first developed as a package. Several options are available: formulation of the fluxes (LP82 or COARE3) or choice of the advection scheme (flux limited vs centered differences). It can be used either for regional modeling purposes (open boundary conditions) or for global modeling (zonally periodic boundary conditions). The current version of CheapAML assumes the model domain is bounded by constant grid lines (e.g., for a sphere, the boundary consists of one northern and one southern latitude, and single eastern and western longitudes).
Acknowledgments
We thank Jean-Michel Campin, Chris Hill, and Thierry Penduff for their valuable help. The final version has benefited from detailed comments by Richard Seager and one anonymous reviewer. This work was supported by NSF Grant OCE-0960500.
APPENDIX
The Turbulent Air–Sea Fluxes
Figure A1 is an example of the differences observed in the strength of the latent and sensible heat fluxes when computed using several methods. Several studies already mention the differences between these products [cf. Kubota et al. (2008) and references therein for an example over the Kuroshio Extension or Rouault et al. (2003) for the Aghulas Current or Kubota et al. (2003) for a global comparison].
Comparison of (left) sensible and (right) latent heat fluxes computed using different bulk formulas in January 2007. (from top to bottom) ERA-40 values, Beljaars (1995); NCEP–NCAR reanalysis; COARE3 using ERA-40 atmospheric and oceanic fields; and LP82 using ERA-40 atmospheric and oceanic fields. Units are W m−2.
Citation: Monthly Weather Review 141, 2; 10.1175/MWR-D-11-00254.1
We report four different computations of the turbulent heat fluxes for January 2007: the ERA-40, Beljaars (1995); the NCEP–NCAR reanalysis, Kalnay et al. (1996); COARE3 using ERA-40 temperature, humidity, and wind; and LP82 also using ERA-40 surface variables.
We observe large differences especially for the sensible heat flux estimations. The latent and sensible heat fluxes are maximum when estimated with LP82. They reach 670 and 250 W m−2, respectively. The patterns are the same for three computations that use ERA-40. It reflects the presence of meanders in the GS. The coarse resolution of the NCEP–NCAR reanalysis does not allow a fine comparison. However, we clearly see that there is a good agreement in the magnitude of NCEP and COARE3.
The comparison of SH-ERA40 and SH-COARE3 is consistent with Fig. 4. It appears that ERA-40 produces significantly lower sensible heat fluxes than COARE3. This difference of almost 100 W m−2 over the warm core of the Gulf Stream can have tremendous effects on the oceanic circulation as illustrated in Fig. 5. Since all atmospheric and oceanic variables are the same in that case in this computation, this difference is only due to the estimation of the exchange coefficient Cd [see (14)].
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