Vertical grid refinement for stratocumulus clouds in the radiation scheme of a global climate model

In this study, we implement a vertical grid refinement scheme in the radiation routine of the global aerosol-climate model ECHAM-HAM, aiming to improve the representation of stratocumulus clouds and address the underestimation of their cloud cover. The scheme is based on a reconstruction of the temperature inversion as a physical constraint for the cloud top. On the refined grid, the boundary layer and the free troposphere are separated and the cloud’s layer is made thinner. The cloud cover is re-calculated either by conserving the cloud volume (SC-VOLUME) or by using the Sundqvist cloud cover routine on 5 the new grid representation (SC-SUND). In global climate simulations, we find that the SC-VOLUME approach is inadequate, as in most cases there is a mismatch between the layer of the inversion and of the stratocumulus cloud, which prevents its application and is itself likely caused by too-low vertical resolution. With the SC-SUND approach, the possibility for new clouds to be formed on the refined grid results in a large increase in mean total cloud cover in stratocumulus regions. In both cases, however, the changes exerted in the radiation routine are too weak to produce a significant improvement of the simulated 10 stratocumulus cloud cover. The grid refinement scheme could be used more effectively for this purpose if implemented directly in the model’s cloud microphysics and cloud cover routines.

clouds, such as stratocumuli, and their radiative fluxes. Although their in-cloud properties and hence cloud optical depth are correctly estimated, forcing them to occupy the thickness of the model layer results in an underestimation of horizontal extent -of cloud area fraction -which alters the all-sky radiative flux.
In this study, we develop and implement a new simple parametrisation for stratocumulus cloud cover in ECHAM-HAM.
We use the inversion reconstruction from Grenier and Bretherton (2001) to define a refinement of the vertical levels in a way that facilitates a more realistic representation of the horizontal extent of simulated stratocumulus clouds. We do not set out to implement the full PBL parametrisation, and instead focus primarily on the stratocumulus cloud cover, using the exact vertical location of the reconstructed inversion as a physical constraint for the cloud top. The number of vertical levels does not increase with our approach as the grid boundary atop the cloudy stratocumulus gridbox is shifted to the inversion pressure, producing a thinner gridbox that matches the true vertical extent of the stratocumulus cloud. In one version of our scheme, we rely on conservation of cloud volume to correct the cloud's horizontal extent, i.e. the cloud cover. In the other, we re-calculate the 70 cloud cover using the cover routine applied on the inversion-based refined grid and profile representations. In order to avoid many numerical problems or difficulties associated with the use of a different grid, we use the new grid and stratocumulus representation only in the radiation scheme of ECHAM-HAM. The radiative effect of stratocumulus clouds is important for climate on a global scale, and hence we hope that the resulting change in radiative transfer and feedbacks can produce an improvement in the simulated stratocumulus clouds overall. 75 In this article we discuss the implementation and results of our stratocumulus cloud cover parametrisation in ECHAM-HAM.
We describe the new scheme's procedure and details of its implementation in Sect. 2 after giving an overview of the model's current treatment of stratocumulus clouds. In Sect. 3.1, we present the results from a test case in single column model (SCM) mode. In Sect. 3.2 and 3.3, we present the results from global climate simulations and discuss the limitations of the scheme's implementation. Finally, we draw our conclusions in Sect. 4.  (Tegen et al., 2019) used with the P3 microphysics scheme developed by Dietlicher et al. 85 (2018). The horizontal resolution is T63 (1.875 • × 1.875 • ) and the vertical is L47 (47 hybrid sigma-pressure levels). The timestep is 450 s and the radiation routine is ran at 'radiation timesteps' i.e. every 7200 s.
For clouds, ECHAM-HAM-P3 uses a two-moment cloud microphysics scheme with one category for cloud droplets and one for ice, and diagnostic parametrisations for rain. Water vapour, liquid and ice are prognostic variables and the cloud cover is diagnosed. ECHAM-HAM's cloud cover scheme is based on the formulation by Sundqvist et al. (1989). Since absolute 90 humidity in the atmosphere varies on scales smaller than the model's gridboxes, subgrid-scale variations in relative humidity (RH) must be parametrised, in order to achieve the formation of clouds in part of the gridbox. The fraction of a gridbox occupied by clouds is named the fractional cloud cover (clc). Given the assumed presence of subgrid variations, clouds must start to form when the gridbox mean RH crosses a threshold value RH c smaller than the saturation relative humidity, RH s = 1.
When the threshold is exceeded, the fractional cloud cover is diagnosed according to Sundqvist et al. (1989): Under low-level inversions, the formula uses adapted parameters (lower RH c and RH s ) with the aim to facilitate the formation of stratocumulus clouds (Mauritsen et al., 2019).

Scheme description
Because the temperature inversion stops vertical motion at the top of the marine boundary layer, we can equate the inversion 100 with the cloud top, and hence use it to constrain the cloud's position and vertical extent. The cloud cover given by the model is the volume fraction that the cloud occupies in the layer. By conserving the cloud volume and restricting the cloud to be found only below the inversion, we reduce its vertical extent and hence increase the horizontal cloud cover, resulting in a more realistic representation of the stratocumulus clouds. The idea is illustrated with a schematic in Fig. 1. This new grid refinement scheme is called invgrid. In the following, we will indicate full-levels (model layers or gridboxes) with an integer index and 105 half-levels (grid boundaries) with half-integers, increasing in the downward direction.
An outline of the method is as follows. First, model columns in which a stratocumulus cloud may be present are identified, and the gridbox layer within which the inversion would be found, named the ambiguous layer, is selected. The exact location (pressure level) of the inversion is diagnosed using the 'reconstructed inversion' method described by Grenier and Bretherton (2001), which assumes a certain sub-grid shape of the temperature profile. The inversion is modelled as a discontinuity in the 110 profile, so that it has a single exact pressure value, usable as the cloud top. Once the inversion pressure is known, the overlying model half-level (representing the cloud top) is shifted down to it, resulting in a thinner lower layer in which the stratocumulus cloud is contained, and a larger but cloud-free above-cloud layer of free tropospheric air. The values of the relevant physical 4 https://doi.org/10.5194/gmd-2020-384 Preprint. Discussion started: 5 January 2021 c Author(s) 2021. CC BY 4.0 License.
quantities are finally recalculated on this new grid. The new grid boundaries and recalculated quantities are passed to the radiation routine, and the procedure is repeated at every radiation timestep.

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The following sections describe the steps in detail, from the detection of applicable columns to the recalculation of all newgrid quantities. The method to calculate the inversion pressure, described in Sect. 2.2.2, follows closely the 'reconstructed inversion' method developed and described by Grenier and Bretherton (2001). The code to calculate the inversion pressure following this procedure was written by Siegenthaler-Le Drian (2010) for her PhD thesis and hence already available in ECHAM-HAM.
A few changes implemented during this study are described.

Ambiguous layer selection
The criterion used to select columns in which to apply invgrid at each timestep is based on low tropospheric stability (LTS).
LTS is a measure defined as the difference in potential temperature between the 700 hPa level and the surface. A strong correlation between LTS and low stratiform cloud cover has been found in observations, especially in the subtropics, as shown by e.g. Klein and Hartmann (1993); Wood and Hartmann (2006). A high LTS is attributable to a strong inversion. Based 125 on the climatology of low stratus cover in Klein and Hartmann (1993), a threshold value of 20 K of LTS is used to select the columns with possible stratocumulus clouds, in which to apply the invgrid scheme. This criterion was previously used by Siegenthaler-Le Drian (2010) to select columns in which to activate her stratocumulus-entrainment parametrisation. As a possible alternative, the threshold could also be based on the estimated inversion strength (EIS), which, as a more refined measure of inversion strength compared to LTS, may be more robust as a predictor of low stratocumulus cloud cover. This is 130 also because, as pointed out by Wood and Bretherton (2006), the relationship between LTS and cloud fraction is not proven to hold in a warming climate, while the link between EIS and stratocumulus cloud cover is more direct. In the context of this study, the choice of criterion between LTS or EIS is not expected to produce significant differences in the selection of stratocumulus columns, and hence the simpler option is used.
In each identified column, the layer in which to look for and reconstruct the inversion must be selected. It is called the 135 "ambiguous layer" by Grenier and Bretherton (2001) because while in reality it would exhibit a lower cloudy part of boundary layer air and an upper cloud-free part of free-tropospheric air, within one model gridbox this vertical distinction cannot be resolved. Finding the inversion pressure allows to separate the two parts. To select the ambiguous layer, we first look for the inversion in the model, i.e. the maximum gradient of temperature. This will be found across two grid layers, which may both potentially contain the inversion jump in a sub-grid profile reconstruction. We choose the ambiguous layer as the uppermost 140 of the two possible layer which contains a cloud, defined as non-zero cloud cover and liquid water content (as in the absence of either of these a cloudy radiative flux is not computed in the model). This selection criterion finds the top of the simulated cloud and hence guarantees that the cloud-rescaling idea would be applicable. If no cloud is present in either of the two possible layers, we use the condition previously used by Siegenthaler-Le Drian (2010): we look at the saturation of an air parcel in an adiabatic ascent from two layers below the inversion, and we choose the ambiguous layer as the layer under the first half-level 145 at which the parcel reaches supersaturation, as it presents the conditions to contain a cloud. This condition operates under the assumption that the stratocumulus is contained within only one layer, as the cloud top could still be found in the layer above.
https://doi.org/10.5194/gmd-2020-384 Preprint. Discussion started: 5 January 2021 c Author(s) 2021. CC BY 4.0 License. The scheme allows the possibility to re-attempt the inversion reconstruction calculation one layer above if it fails in the first selected ambiguous layer.

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We diagnose the inversion pressure following the method developed and described by Grenier and Bretherton (2001); the procedure is repeated here for convenience of the reader and to indicate our modifications.
The inversion pressure reconstruction method by Grenier and Bretherton (2001) is based on reconstructing the sub-grid profile of virtual liquid water potential temperature (θ vl ) in the ambiguous layer k, in which its value θ k vl is considered the weighted average of its below-inversion (boundary layer) and above-inversion (free-tropospheric) values. Figure 2 shows a 155 diagram of an example profile, with labelled layer indices.
where T is temperature, p is pressure (p 0 = 1000 hPa), R d and R v are the dry air and water vapour gas constants, L v is the latent heat of vaporisation, q l and q t are the liquid and total water contents (mass mixing ratios), and c pd is the constant 160 pressure heat capacity of air. θ vl depends linearly on temperature, and hence exhibits the same vertical profile features (notably the inversion), with the advantage of being a conserved quantity in a reversible moist adiabatic process, i.e. a process where all the condensate remains within the air parcel. The inclusion of the 'potential' and 'liquid water' part (second and third factors on the right hand side of the equation) causes the quantity to be conserved, while the 'virtual' part (fourth factor) allows to use the dry-air equation of state. This makes the quantity advantageous to use in calculations, and hence θ vl is used for the profile 165 reconstruction.
On a sub-grid scale, within the ambiguous layer, we distinguish between an above-inversion and a below-inversion profile, and assume that θ vl follows Below the inversion, θ vl has the same value that it does lower down in level k + 1, justified by the fact that in the case of a 170 strong inversion, the boundary layer tends to be very well-mixed, in which case θ vl is constant throughout the well-mixed layer.
Above the inversion, the θ vl profile is extrapolated down from the overlying level, using the maximum negative gradient with respect to pressure (s) chosen from the gradients across half-levels k − 1/2 or k − 3/2.
This profile implies that at the inversion pressure p inv , θ vl experiences a discontinuity, where the value jumps from the boundary layer to the free troposphere one, representing the sharp inversion. The inversion pressure is found by requiring 175 conservation of θ vl within the ambiguous layer, i.e. by requiring that the integral of the sub-grid profile is equal to the original value of θ vl in the ambiguous layer k (θ k vl , considered to be the gridbox average): In order to solve Eq. (5) for the inversion pressure, we define the above-inversion mass fraction of the ambiguous layer, Equation (5) can then be turned into a quadratic equation in µ: The physical solution for µ is a value between 0 and 1 which, when it exists, can be shown to be the smaller solution of Eq. (7).
If and when a physical µ is found, the inversion pressure p inv is obtained inversely from Eq. (6). In a following step, p inv is 185 used to define the new grid.
less ideal, situations. For example, we force a small but negative s (−1 × 10 −6 K Pa −1 ) if the gradient above the inversion is only slightly positive (which would normally be considered unusable). We also attempt to carry out the inversion reconstruction in the upper possible ambiguous layer if it fails in the lower one.

Grid refinement
As the new representation is used exclusively in the radiation routine, the grid refinement is applied only in cases where it 195 would make a difference to the radiative transfer calculations, specifically by increasing the cloud cover. Hence, we first check that the ambiguous layer contains a cloud, as this is a necessary condition for the radiation routine to compute a cloudy flux.
We also ensure that the gridbox layer would not become thinner than a minimum thickness. The limit is put in place to prevent unphysical situations, such as for example a too-high liquid water mixing ratio or cloud droplet concentration. We choose a threshold of 50 m, as stratocumuli are almost never observed to be thinner according to cloud thickness climatologies such as 200 in Wood (2012).
If the conditions are appropriate, we proceed with defining the new refined grid. The half-level above the inversion, the top of the ambiguous layer, is shifted down to the inversion pressure p inv . Level k becomes thinner, and will wholly contain the cloud that was originally present in the ambiguous layer. Level k − 1 on the other hand becomes larger, and will represent the first layer of free tropospheric air.

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Once the new grid is defined, the variables that need to be passed to the radiation routine are calculated in the new layers using the assumed sub-grid profile, conservation principles, and the notion that the stratocumulus in the new grid is constrained below the inversion. The procedure for each variable is detailed in the following. Superscripts k and k − 1 refer to variables and layers in the original model grid, i.e. the ambiguous layer and the overlying layer respectively. We use superscripts kinv for the new thinner layer (equivalent to the below-inversion fraction of the ambiguous layer), abinv for the above-inversion 210 fraction of the ambiguous layer (note that this is not a layer in its own right on either grid), and kinv − 1 for the new larger overlying layer, consisting of layers abinv and k − 1.

Water content reconstruction
The water vapour q v , liquid water q l and ice q i contents are defined as mass mixing ratios (kg kg −1 air ) in the model. The total water content q t is the sum of all three individual phases and must be conserved.
For consistency with θ vl , we require that q t follows the same sub-grid profile in the ambiguous layer, and start by calculating it as where the second equation is a solution to conservation of q t in layer k, given the above-inversion mass fraction µ of the 220 ambiguous layer. Its value in kinv − 1 is obtained as a mass-weighted average of abinv and k − 1: where M is the air mass of a layer and the denominator is equal to the air mass in kinv − 1.
Since liquid water and ice are the components that make up the cloud, we restrict them to be found only below the inversion, in the new thinner cloud layer kinv. The total liquid and ice mass is conserved, which means the mixing ratio is simply rescaled 225 to the new layer mass: The quantities q l and q i are thus assumed to be zero in abinv, and for layer kinv − 1 the values in k − 1 are rescaled to the larger layer. This recalculation does not change the in-cloud values of q l and q i , which are used in the radiation routine, as the cloud volume is conserved.

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The calculation of the water vapour content in the new layers kinv and kinv −1 uses the previously calculated reconstructed total water and the inversion-constrained liquid and ice water contents: We recognise that the method for recalculating the water contents described in this section is not fully consistent. In fact, the total water content is treated as a separate variable (as opposed to using the sum of the individual phases of water) and is 235 reconstructed using the sub-grid profile which depends on the under-and overlying layers; at the same time the liquid and ice contents are taken from the ambiguous layer and simply moved to the below-inversion part. This can lead to an inconsistency in the water vapour content especially in the new cloudy layer kinv (in which the air should be saturated), as it is calculated by subtracting the rescaled liquid and ice contents (from layer k) from the total q t (from layer k +1). In kinv −1 the inconsistency is negligible as there should be no liquid or ice water there. We decided to move forward with this method despite this problem 240 because it has the following advantages. With this method, the liquid and ice contents used for the cloud are the ones that are calculated in the cloud microphysics routine, which takes into account fundamental microphysical processes. Also, the resulting total water below the inversion is equal to the one in the layer below, as is characteristic of well-mixed boundary layers. Overall, the method gives reasonable results for q v above the inversion and for q l and q i in the cloudy layer below it, and the total water content as sum of the individual components is indeed conserved. Checks are in place to prevent and fix 245 potential unphysical (negative) values of q v , q l , q i or q t .

Temperature
The temperature on the new grid is calculated using energy conservation. First, in kinv, T is obtained inversely from θ kinv vl (Eq. (2)). Then, the heat content H of the original layers k and k − 1 is calculated, along with the heat content of new layer kinv: where superscript j indicates the layer considered and c pv , c lw and c iw are the vapour, liquid water and ice specific heat capacities. Then, for energy conservation over the two layers between the original and new grid, T kinv−1 is obtained from Further cloud variables 255 Similar to q l and q i , we confine all cloud variables of the ambiguous layer k to the new thinner layer kinv, which is capped by the inversion. The recalculation invokes conservation of cloud volume for cloud cover (clc), and particle number for cloud droplet and ice crystal number concentrations (n cd , n ic ). The in-cloud concentrations hence also remain constant. The variables are simply scaled to the new layer thickness Z, essentially 'squeezed' under the inversion: The cloud cover is of course constrained not to exceed 100%.
The new half-and full-level pressures (grid boundaries) and all the recalculated new-grid variables are finally passed to the radiation routine.

Model versions 265
The model versions that were used to perform the simulations discussed in the next sections are presented in the following.
In addition to the reference model version (REF), two versions implementing invgrid were used: one which rescales cloud cover based on cloud volume conservation, as described above, (SC-VOLUME); one which re-calculates the cloud cover on the refined grid by re-running the model's Sundqvist cloud cover scheme (SC-SUND). Another simple scheme (SC-MAX) was used to test and provide an understanding of the potential and limitations of the different invgrid versions.

SC-VOLUME
In the SC-VOLUME model version, the invgrid scheme described in Sect. 2.2 is fully implemented in the model. The calculation of the inversion pressure as in Sect. 2.2.2 is performed at every timestep before the radiation routine for diagnostic reasons. At radiation timesteps, the value is used to refine the vertical grid; physical variables are recalculated as described in Sect. 2.2.3, with the stratocumulus cloud cover calculation being based on cloud volume conservation. These are passed to the 280 radiation routine.

SC-SUND
In the SC-SUND model version, after executing the invgrid grid refinement, the stratocumulus cloud cover is calculated by running the model's Sundqvist cloud cover scheme. The goal is to address cases in which, on the original grid, no cloud is present in the ambiguous layer. This could be due to the ambiguous layer's water vapour mixing ratio being an average between 285 dry tropospheric air and moist boundary layer air, which may cause the gridbox average relative humidity to be too low to reach the threshold for forming cloud cover according to Eq. (1). With the new grid's reconstruction, the two different air masses are separated, which may allow a cloud to form in the new thinner layer, now made up exclusively of boundary layer air and hence presumably having a higher relative humidity. This would be valuable because it would lead to a better representation of stratocumulus clouds in layers in which the method of SC-VOLUME could not be applied due to the lack of a cloud in the 290 model in the first place. This method makes use of the refined grid and recalculated profiles of water content and temperature, but the cloud volume is not necessarily conserved as the cloud cover is recomputed with the cloud cover scheme. The procedure is only applied if the layer in which a new cloud cover is calculated contains liquid water (or cloud ice). The new cloud cover representation is only used in the radiation routine.

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The SC-MAX model version was designed to investigate the maximum possible effect of a scheme that increases the cloud cover of existing stratocumuli, such as in SC-VOLUME. This is done by always increasing the cloud cover to 100% in model layers where a stratocumulus cloud is identified. The cloud cover increase is applied in the same cases in which SC-VOLUME's cloud rescaling would be, i.e. when the identified ambiguous layer contains a cloud, but also when the ambiguous layer contains no cloud but another layer, at most two levels below it, does. We still consider the latter case as a stratocumulus cloud. The 300 cloud cover of the first (uppermost) cloudy model layer below the inversion is set to 100%. The modified cloud cover is passed to the radiation routine.

Experiment description
For each model version, we performed a 15-year-long (2000-2014) global climate simulation with prescribed AMIP seasurface temperatures (PCMDI, 2018) to evaluate the stratocumulus cloud representation. We used the standard ECHAM-HAM 305 T63/L47 spatial resolution and 450 s (7.5 minute) timestep. The data from the invgrid routine is sampled at radiation timesteps, i.e. every 2 hours. As an observational reference for total cloud cover, we used Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (Calipso) data from the GCM-Oriented Calipso Cloud Product (GOCCP) dataset (Chepfer et al., 2010).
3 Results and discussion

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We first tested invgrid's inversion reconstruction and grid refinement in ECHAM-HAM's single column model (SCM) mode (Dietlicher et al., 2018). Using the SCM allows us to closely observe the evolution of the vertical profiles and how invgrid responds to them. Additionally, the possibility to use observational forcings for the SCM is a method to test the model's representation of real situations and to generally validate the reconstructions of the new scheme. The validation in the SCM was carried out using a forcing derived from observations made during the EPIC campaign (Bretherton et al., 2004), specif-315 ically from a segment between 16-22 October 2001 in the southeastern Pacific, where the vertical structure of the boundary layer capped by a persistent stratocumulus cloud was observed using radiosondes and remote sensing (Bretherton, 2005). The EPIC campaign also provided observations of the cloud top and base in this period, obtained with cloud radar and ceilometer respectively (Caldwell et al., 2005), which are used to validate the found inversion heights. Figure 3 shows the evolution of the cloud top and base over the 6-day period of the EPIC campaign. The cloud top follows the 320 PBL's diurnal cycle, rising during the night due to longwave cloud top cooling driving entrainment and sinking during the day due to absorption of solar radiation suppressing entrainment. The reconstructed inversion generally captures this diurnal cycle.
While the exact height of the inversion is at times overestimated (days 1, 4), most of the time it matches the observed cloud top quite well, especially on days 2, 3 and 6. The occasional sudden jumps of the inversion pressure (e.g. between days 1 and 2) occur due to the selection criterion for the ambiguous layer depending on the maximum gradient of θ vl , whose level can change 325 suddenly when the inversion is not very sharp. Finding a criterion which could address this undesirable issue without loss of generality proved difficult. We also calculated the lifting condensation level (LCL) for an air parcel rising from the surface to attempt to estimate the cloud base, but the results exhibited large oscillations and did not match the cloud base most of the time, rendering the LCL diagnostic unsuitable as a proxy for cloud base. A reconstruction of the cloud base would be beneficial to the scheme to complement the constraint of the cloud's extent from above with a constraint from below, resulting in a further    in cases where the method previously failed, for example at the start of the first day or during day 6. Overall, the inversion reconstruction method gives good results for finding the location of the stratocumulus cloud top. 335 We show some example vertical profiles which occurred during EPIC to illustrate the effect of the grid refinement on temperature, total water mixing ratio and cloud cover in Fig. 4. The inversion in the physical quantities is sharper, and demonstrates the better separation between the PBL and the free troposphere on the refined grid. The cloud cover increases by 6 percentage points -or by almost 30% -as a result of the lower vertical extent of the layer.
cloud cover below 800 hPa for each experiment. Only the cloud cover belonging to layers which also have a non-zero liquid water content is shown. In the REF simulation, the cloud is mostly contained within one layer in the model. In fact, it is found most often in the layer below the one containing the inversion, and hence in which cloud top was predicted (and observed) to be. In these situations, with SC-VOLUME the cloud rescaling cannot be applied, as the layer containing the inversion contains no cloud, and hence the reconstruction would not make a difference to the radiation routine. The three times where the 345 model's cloud extends into the upper layer, the invgrid scheme is applied effectively, reducing the thickness of the top cloudy layer following the inversion and hence obtaining a more realistic depiction. The SC-SUND scheme version was developed in response to the issue described above: it uses a better water profile representation by applying the refined grid also in the cloud cover routine, so that possibly a new cloud can be formed right below the inversion that is missing when using the original grid.
In the SC-SUND simulation, a new cloud is formed in a few cases in the upper layer (days 1, 2, 4) and once in the central layer 350 at the end of day 5. The scheme actually simulates a new cloud cover more frequently, but the lack of water condensate in the inversion layer limits the number of 'valid' instances. A more ideal representation of the water content in addition to clc would be obtained if the grid refinement method were used in the microphysics routines too. Such an implementation comes with the aforementioned challenges that our more limited usage of the new scheme in the radiation routine only aimed to avoid.
In our analysis of the global simulations we also investigate the frequency of situations such as those observed in the 355 EPIC simulations, in which the model's simulated cloud is in the layer below where we expect to find it via the inversion reconstruction.

Global Climate Simulations
After demonstrating the desired functioning of invgrid in the SCM, we studied its effect on the stratocumulus cloud cover in global climate simulations. We focus on three subtropical stratocumulus regions, which are known to exhibit semi-permanent 360 marine stratocumulus sheets, namely the oceans just west of North America (NAM), South America (SAM) and southern Africa (AFR). We also look at an Arctic region, over the Barents sea (BAR). The regional averages cited in the text are defined over the areas highlighted in Fig. 6a.
The reference model version REF generally underestimates cloud cover in the subtropical stratocumulus regions, as shown in Fig. 6b in a comparison to the Calipso-GOCCP satellite climatology (Chepfer et al., 2010). The cloud cover difference 365 exhibits a similar pattern in all three regions: compared to observations, cloud cover is actually overestimated along the coast, such that the overall underestimation results from large areas of lower cloud cover further offshore. In the Arctic, total cloud cover is instead overestimated by the model.
The results from the simulations with the modified schemes are shown in Fig. 6c-h, with on the right hand side the annual mean simulated total cloud cover, and on the left hand side the total cloud cover change experienced by the radiation routine, i.e. the difference between after and before the application of the invgrid scheme. The total cloud cover in the simulations can change when changes by the invgrid scheme on cloud radiative effects feed back on the clouds, e.g. by increased turbulence through stronger cloud top cooling. Regional averages are reported in Table 1.   with Calipso climatology. Results with SC-VOLUME, SC-SUND, SC-MAX: (c)-(h) on the left, total cloud cover increase exerted in the radiation routine; on the right, change in simulated total cloud cover compared to REF (stippling indicates statistically significant differences at the 95% significance level; the false discovery rate is controlled following Wilks (2017)). Fig. 6a are defined as follows: NAM:

The regions highlighted in
In the SC-VOLUME simulation, the increase in total cloud cover caused by invgrid and seen by radiation in the annual mean is extremely small in stratocumulus regions, reaching at most 1 percentage point (pp) in the Arctic where it is most 375 marked. As the changes in the radiation routine are small, the change induced to the simulated cloud cover due to internal climate feedbacks is also very small. A simple two-sided t-test using the annual means showed that the results do not differ from REF in a statistically significant manner; they also do not exhibit an explicable pattern (Fig. 6d). The changes in cloud radiative effects produced with SC-VOLUME were much weaker than we had initially expected (not shown), and in Sect. 3.3 we investigate the factors that limit the effectiveness of the SC-VOLUME method in global simulations. Table 1. Annual mean total cloud cover and differences between simulations, as global and regional averages. An asterisk denotes statistically significant differences at the 95% significance level. "∆ seen by rad. " indicates the change in total cloud cover produced with invgrid, which is then applied only in the radiation routine. In the SC-SUND simulation, the possibility to form new clouds on the refined grid gives the potential to produce a larger mean cloud cover increase than with SC-VOLUME. This is in fact the case in the radiation routine (Fig. 6e): as intended, the subtropical stratocumulus regions exhibit large increases (up to 15 pp) in the annual mean total cloud cover. The most affected areas are located away from the continental coasts, i.e. in the regions where ECHAM-HAM most underestimates cloud cover, showing that SC-SUND can accurately address the problem. As for the change induced in the simulated total cloud cover 385 (Fig. 6f), the difference to REF is also small (on the same order of magnitude as with SC-VOLUME), although the spatial patterns seem to indicate a slight reduction of the model bias in subtropical stratocumulus regions. The stronger cloud top radiative cooling could favour convection bringing moisture into the cloud from the surface, increasing stratocumulus cloud longevity in a positive feedback loop affecting the simulated cloud cover. However, the difference with REF was found to be not statistically significant with SC-SUND as well.

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With both set-ups, the change exterted is too small to cause significant changes in the simulated total cloud cover. At the same time, the results indicate that, in terms of the initial changes produced in the radiation routine, SC-SUND is more effective than SC-VOLUME at increasing cloud cover in the annual mean. This suggests that the model's bias is less due to an underestimation of cloud extent in individual instances, which SC-VOLUME is designed to address, and more to a negative bias in the frequency of stratocumulus cloud formation, which can be addressed by SC-SUND due to its re-evaluation of cloud 395 cover on the refined grid. Other factors hindering the suitability of SC-VOLUME could also be at play, and they are considered in the following Section.

Further analysis and scheme limitations
3.3.1 Scheme usage frequency in SC-VOLUME For SC-VOLUME's cloud cover reconstruction to produce a significant effect in global simulations, it must be applied fre-400 quently in practice. The invgrid scheme requires a series of conditions to be met in order to be applied and rescale the ambiguous layer's cloud cover. The occurrence frequency of these conditions in the SC-VOLUME simulation is reported in Table   2.
First of all, the scheme must find and successfully reconstruct a temperature inversion. The associated conditions and calculations, described in Sections 2.2.1 and 2.2.2, result in the inversion being found very frequently in the stratocumulus regions, 405 upwards of 70% in some columns (Figure 7a). In most of these cases, a stratocumulus cloud, defined as a cloudy layer at or below the inversion, is also present (Figure 7b). The occurrence frequency of these identified stratocumulus clouds is lower than in reality, where it is around 46% annually in the relevant regions according to the ship-based observational climatology  by Hahn and Warren (2007). This represents a deficiency of the model and a limitation to the SC-VOLUME scheme's aptness to correct the cloud cover bias. The practical applicability of SC-VOLUME's cloud reconstruction method 410 is even more starkly reduced by the subsequent necessary condition, that the cloud (or at least its upper part) must be found in the same model layer as the inversion. As indicated in Sect. 3.1 and quantified in Fig. 7c, this condition is in fact very rare and occurs in much more limited areas than those in which stratocumulus clouds are identified, concentrated in close proximity of the coasts. Figure 7d shows the conditional probability of the stratocumulus cloud being found in the ambiguous layer, given that a cloud is present below the inversion. This probability decreases with the distance from the coast in the subtropical marine 415 stratocumulus regions, where overall it is less than 25%. The rest of the time, the cloud is at a lower level than the inversion.
The conditional probability is instead very high in higher latitudes. This is likely the result of the different meteorological conditions -due to lower temperatures and the presence of ice, the model's RH requirement for cloud cover formation is lower and easier to reach. In addition, the PBL is typically shallower in the Arctic, and as the model's vertical resolution is higher closer to the surface, its vertical structure is better resolved and can more easily form clouds at the right level.

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The results indicate that there is a prevalent mismatch between the layer where the inversion is found, which is where the cloud top is expected to be, and where the model in fact forms the cloud, in the layer below the inversion. In these cases, the idea of 'squeezing' the existing cloud under the inversion cannot be used, and hence this discrepancy between predicted and effective location of the cloud in the model greatly reduces the applicability of SC-VOLUME's method, especially in the subtropics. cloud is present below the inversion, (c) a cloud is present in the ambiguous layer, (d) given that there is a cloud below the inversion, the cloud is in the ambiguous layer (conditional). Table 2. Global and regional average occurrence frequency in the SC-VOLUME simulation of finding an inversion, finding an inversion with an underlying cloud (identified stratocumulus clouds), finding a cloud in the ambiguous layer (AL); and the conditional occurrence frequency of a cloud in the ambiguous layer, given that a cloud is present below the inversion. Also included for comparison are the average frequency of occurrence of stratocumulus clouds from the Hahn and Warren (2007)   the PBL air, and therefore will be unlikely in the ambiguous layer especially when the inversion is close to its bottom. The layer below the ambiguous layer, being fully located inside the PBL, is instead much more likely to present the conditions appropriate to form a cloud in the model. Thus the stratocumulus cloud is most often found in the layer below the ambiguous layer, rather than in the ambiguous layer itself. The problem we identified of misrepresentation of the cloud's vertical location 435 seems to be a result of poor vertical resolution, just like the underestimation of stratocumulus cloud cover due to exaggeration of their vertical extent. Our scheme aimed to correct the latter, but doing so as we envisioned is difficult without also addressing the former.

Maximum cloud cover improvement with SC-VOLUME
In addition to only being used in a small fraction of stratocumulus cases, we found that SC-VOLUME's cloud reconstruction 440 does not tend to increase cloud cover very much in the layers in which it is used. Figure 8a shows the mean cloud cover in the ambiguous layer when it contains a cloud. In the stratocumulus regions, close to the coasts the ambiguous layer cloud cover is already very high on average, and hence cannot be increased much further, but farther offshore it decreases as low as 40 %. However, the mean increase produced there is less than 10 pp (Fig. 8b). A probable reason for this is that, when inversion and cloud layer match, the inversion is likely to be high within the layer (as it is the associated higher proportion of PBL 445 in the layer that allowed the formation of a cloud). Hence, the refined layer is not much thinner than the original one, and a volume-conservation-based reconstruction of the cloud cover does not increase it very much. This demonstrates again how the SC-VOLUME method for cloud cover reconstruction is limited by the same biases of the original vertical representation that invgrid aims to correct. While the grid refinement can improve the vertical representation, basing the new cloud cover on the flawed original cloud cover gives poor results.

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To assess the maximum effect that a scheme such as invgrid in SC-VOLUME, increasing the cloud cover for existing stratocumulus clouds, could cause, we performed the SC-MAX experiment. In this simulation, the cloud cover of identified stratocumulus clouds, i.e. the first cloudy layer at or below the inversion level, is set to 100%. The SC-MAX method is applied also to those stratocumulus situations that SC-VOLUME could not affect (in which the cloud and the inversion are not in the same layer). Hence, the SC-MAX method exerts the maximum possible stratocumulus cloud cover increase. Table 3. Global and regional averages of mean ambiguous layer cloud cover (clc kinv ) and its increase in SC-VOLUME, conditionally sampling cases in which the ambiguous layer contained a cloud. The annual mean total cloud cover difference that is produced for the radiation routine with SC-MAX is very large, as can be seen in Figure 6g. Further, in this case the changes exerted propagate through feedbacks much more evidently, and can be clearly observed in the model's simulated total cloud cover. When comparing it to REF (Fig. 6h), the increase exhibited in the subtropical stratocumulus regions is significant. However, the model's bias compared to observations is still far from being completely corrected, as the average underestimation in the South American region, which experienced the most improvement, 460 is still -11.7 pp with SC-MAX as opposed to -13.5 pp in REF.
The SC-MAX experiment demonstrates that a stratocumulus cloud cover scheme applied only in the radiation routine can have a positive effect on the model via feedbacks, but, in the case of ECHAM-HAM, it is not sufficient to fully close the gap between the simulated and observed cloud cover. Even the cloud cover seen by the radiation routine is still underestimated in the stratocumulus regions compared to the observed climatology. This experiment further confirms that ECHAM-HAM's 465 cloud cover bias is caused also by a lack of stratocumulus clouds in the first place. A scheme affecting only existing clouds, such as in SC-VOLUME, would need to be complemented by improvements in other parametrisation schemes to increase the occurrence of stratocumulus clouds as well.
The SC-SUND scheme presents a possible improvement to this, as it can be applied even in columns with no below-inversion cloud at all, with the possibility to form a new cloud there.

New clouds in SC-SUND
The SC-SUND scheme has the potential to address both the issues identified in the previous sections in the SC-VOLUME setup, namely its inability to address cases in which the stratocumulus is below the inversion layer and the scarcity of simulated stratocumulus in the model.  its increase compared to the original representation. The SC-SUND scheme forms new clouds up to 30% of the time in the southern marine subtropical stratocumulus regions, also significantly extending over the ocean the area over which the condition occurs. This demonstrates that the separation of PBL and free troposphere achieved with the refined grid is very effective, and allows clouds that could not be formed previously in the coarser gridbox to be 'revealed' on a re-execution of the cloud cover scheme. The overall effect that a newly formed cloud in the ambiguous layer has on total cloud cover depends on 480 the presence of a cloud in other layers: in a previously cloud-free column a new cloud increases the total cloud cover in general more significantly than in a column already containing a cloud. The total cloud cover change experienced in the annual mean by the radiation routine in the SC-SUND simulation is quite large, reaching up to 15 pp in some columns and affecting extended areas (Fig. 6e). It is of comparable magnitude to that in the SC-MAX simulation (Fig. 6g). However, it is interesting to note that the effect that is then produced on the simulated total 485 cloud cover is much less marked with SC-SUND than with SC-MAX (see Fig. 6f and 6h). The reason for this may be linked to the fact that the process by which the annual mean total cloud cover seen by the radiation routine is increased is different in the two set-ups. In SC-MAX, the annual mean increases because all simulated stratocumulus clouds become 100% covering, but their number remains the same. In SC-SUND, it increases because the scheme can form new clouds and hence stratocumulus clouds appear more frequently, with their coverage being calculated as usual. Thus, in SC-MAX, sunlight is scattered back to 490 space almost fully in all situations with a stratocumulus cloud, which can have a very drastic effect on the radiation balance in stratocumulus regions; on the other hand in SC-SUND the increase in shortwave scattering in stratocumulus regions is more evenly distributed over time and may hence produce a more moderate effect on the meteorological conditions in those regions.
In addition, the effect produced in SC-SUND may be weaker because the new clouds occurring in the scheme, although they have a more realistic cloud cover, are likely to have a too-low liquid or ice content. Their liquid or ice content comes from the 495 condensation or deposition computed using the original grid's gridbox mean RH, i.e. at low supersaturation, or from transportboth resulting in low amounts. This is a disadvantage of SC-SUND, as to have realistic liquid or ice content the grid refinement should be applied in the cloud microphysics scheme too. The water content in the SC-MAX clouds is likely higher, as there RH was high enough to form cloud cover.
These results indicate that the cloud cover improvement obtained in the radiation routine thanks to the implementation 500 of invgrid is 'lost', as the radiative impact is too weak for the changes to be propagated to the simulated climate, probably due to a too-low water content in the new clouds. Despite the higher complexity, it may be beneficial to implement the grid refinement scheme directly in the cloud-related microphysics and cover routines in order to obtain a sizeable improvement in ECHAM-HAM's simulated stratocumulus clouds. Two parametrisations for stratocumulus cloud cover based on a vertical grid refinement at the level of the capping inversion were developed and implemented only in the radiation routine of the ECHAM-HAM GCM. SC-VOLUME uses a geometrical and physical argument to augment the cloud's horizontal extent under the inversion; SC-SUND makes use of the improved temperature and water profile at the inversion and re-evaluates the cloud cover.
The inclusion of SC-VOLUME did not lead to significant improvements for the model's cloud cover bias in long-term 510 global climate simulations. Some limitations of the method were highlighted. Firstly, the simulated stratocumulus clouds are very rarely occurring in the model layer containing the inversion and appear instead more often in an underlying layer, revealing a bias in the model's representation which may be due to a poor resolution of the humidity profile. As the correspondence of inversion layer and cloud layer is a necessary condition for the application of SC-VOLUME's cloud constraining method, this means that the scheme can only be applied in a small fraction of the identified stratocumulus cloud cases, limiting its general 515 effect. Having identified this common stratocumulus-inversion layer mismatch is valuable, as it explains why a geometry-based method for the representation of stratocumulus clouds such as SC-VOLUME is not widely applicable and suggests that it would work better with a higher vertical resolution -which would improve stratocumulus cover representation regardless. Secondly, even in cases in which it is applied, SC-VOLUME tends to produce only a relatively small increase in the layer's cloud cover.
With the SC-MAX experiment we showed that the model's stratocumulus cover bias is not only due to an underestimation 520 of simulated clouds' horizontal extent, but also to an underestimation of their occurrence frequency and the areas where they appear. Hence, we conclude that a method like SC-VOLUME, addressing only pre-existing clouds, is too limited.
The SC-SUND scheme aimed to address both of the issues limiting SC-VOLUME. In fact, its application led to the formation of new clouds in the refined below-inversion grid layers and hence to a larger increase of the total cloud cover seen by the radiation routine compared to SC-VOLUME. Through some feedbacks this positive effect is perceived also by the simulated 525 cloud cover, which increased and slightly reduced the bias to observations -however, this effect remains minor compared to the bias's magnitude. Changes to stratocumulus cloud representation, if applied in the radiation routine only, are hardly passed on to the model's general representation. Therefore, despite its numerical advantages, the implementation of a stratocumulus cloud parametrisation limited to the radiation routine is mostly unprofitable.
As the developed grid refinement method itself works well and improves stratocumulus cloud cover within the radiation 530 routine, where it is currently applied, it could be valuable in the future to expand its use to other parts of the model as well.