# Improving stratocumulus cloud amounts in a 200-m resolution multi-scale
modeling framework through tuning of its interior physics

###### Abstract

High-Resolution Multi-scale Modeling Frameworks (HR) – global climate models
that embed separate, convection-resolving models with high enough resolution
to resolve boundary layer eddies – have exciting potential for investigating
low cloud feedback dynamics due to reduced parameterization and ability for
multidecadal throughput on modern computing hardware. However low clouds in
past HR have suffered a stubborn problem of over-entrainment due to an
uncontrolled source of mixing across the marine subtropical inversion
manifesting as stratocumulus dim biases in present-day climate, limiting their
scientific utility. We report new results showing that this over-entrainment
can be partly offset by using hyperviscosity and cloud droplet sedimentation.
Hyperviscosity damps small-scale momentum fluctuations associated with the
formulation of the momentum solver of the embedded large eddy simulation. By
considering the sedimentation process adjacent to default one-moment
microphysics in HR, condensed phase particles can be removed from the
entrainment zone, which further reduces entrainment efficiency. The result is
an HR that can produce more low clouds with a higher liquid water path and a
reduced stratocumulus dim bias. Associated improvements in the explicitly
simulated sub-cloud eddy spectrum are observed. We report these sensitivities
in multi-week tests and then explore their operational potential alongside
microphysical retuning in decadal simulations at operational 1.5 degree
exterior resolution. The result is a new HR having desired improvements in the
baseline present-day low cloud climatology, and a reduced global mean bias and
root mean squared error of absorbed shortwave radiation. We suggest it should
be promising for examining low cloud feedbacks with minimal approximation.

Journal of Advances in Modeling Earth Systems

Department of Earth System Science, University of California, Irvine,
California, USA Department of Atmospheric Sciences, University of Washington,
Seattle, Washington, USA Lawrence Livermore National Laboratory, Livermore,
California, USA Allen Institute for Artificial Intelligence, Seattle,
Washington, USA NVIDIA, Santa Clara, California, USA

Liran<EMAIL_ADDRESS>
We improve a long-standing stratocumulus (Sc) dim bias in a high-resolution
Multiscale Modeling Framework.

Incorporating intra-CRM hypervisocity hedges against the numerics of its
momentum solver, reducing entrainment vicinity.

Further adding sedimentation boosts Sc brightness close to observed, opening
path to more faithful low cloud feedback analysis.

## Plain Language Summary

Stratocumulus clouds cover a large fraction of the globe but are very
challenging to reproduce in computer simulations of Earth’s atmosphere because
of their unique complexity. Previous studies find the model produces too few
Stratocumulus clouds as we increase the model resolution, which, in theory,
should improve the simulation of important motions for the clouds. This is
because the clouds are exposed to more conditions that make them evaporate
away. On Earth, stratocumulus clouds reflect a lot of sunlight. In the
computer model of Earth, too much sunlight reaches the surface because of too
few stratocumulus clouds, which makes it warmer. This study tests two methods
to thicken Stratocumulus clouds in the computer model Earth. The first method
smooths out some winds, which helps reduce the exposure of clouds to the
conditions that make them evaporate. The second method moves water droplets in
the cloud away from the conditions that would otherwise make them evaporate.
In long simulations, combining these methods helps the model produce thicker
stratocumulus clouds with more water.

##  1 Introduction

Stratocumulus (Sc) clouds play an important role in the Earth’s radiation
budget. They are extensive, long-lived, and cool the Earth by reflecting
downwelling solar radiation back to space while having little impact on the
outgoing longwave radiation. Primarily formed in the presence of large-scale
subsidence over cold oceans, the annual mean Sc coverage over the ocean and
land is $23\%$ and $12\%$, respectively [Wood (2012)]. Sc also has a strong
influence on the heat and moisture exchange between the troposphere and
boundary layer [Randall . (1984)]. Despite the climatic significance of Sc,
climate models do not agree on their seasonal cycle, spatial extent, radiative
properties, and cloud feedbacks [Bony . (2011), Lin . (2014), Gettelman
Sherwood (2016), Brunke . (2019), Vignesh . (2020), Tselioudis . (2021),
Konsta . (2022), Zelinka . (2022)], and even high resolution models simulate
widely varying cloud properties in idealized case studies [Ackerman . (2009),
Stevens . (2005), Bretherton . (1999)]. More realistic simulated Sc is
necessary to improve our understanding of Sc physics and confidence in
projections of the future global-mean temperature [Bony Dufresne (2005), Webb
. (2013), Dal Gesso . (2015), Tsushima . (2016), Schneider . (2017), Zelinka .
(2020)].

Simulating Sc remains a particular challenge because Sc processes involve a
wide range of spatial scales and key physical processes that are poorly
represented in the subgrid-scale parameterization of global models. Although
an Sc cloud deck might cover tens to thousands of kilometers, its thickness is
typically only a few hundred meters [Wood (2012)]. Cloud-aerosol interactions
are modulated through changes in cloud droplet number, which itself depends on
the strength of updrafts whose scale is on the order of 10s or 100s of meters.
At the top of subtropical stratocumulus clouds, intense mixing between warm,
dry free-tropospheric air and the underlying wet cloud layer occurs within a
thin layer typically less than 20 m in vertical extent [Caughey . (1982),
Haman . (2007), Mellado (2017)]. bretherton2015insights suggests that such
cloud-top entrainment plays a leading role in multiple cloud feedback
mechanisms in stratocumulus. Because low cloud feedbacks and cloud-aerosol
interactions in stratocumulus clouds are thought to be controlled in part by
fine-scale processes that are not represented explicitly in storm-resolving
models ($\sim$ 1km), this motivates simulations with sub-kilometer grid
spacing [<]e.g., ¿stevens2020added.

Several interesting strategies have emerged in recent years to capture more
explicit and plausible stratocumulus dynamics in next generation global
climate models. First, lee2022resolving demonstrate some potential from
adaptively refining vertical grid structure solely within a strategic subset
of the physical parameterization suite, with higher order closure scheme used
in Cloud Layers Unified By-Binormals (CLUBB,
golaz2002pdf2,golaz2002pdf,larson2005using,larson2012pdf) and vertical
transport (e.g. subsidence and sedimentation/precipitation). This has
advantages of producing some better baseline Sc [Bogenschutz . (2013), Lee .
(2022)], but disadvantages of accepting all the limitations of operational
subgridscale turbulence schemes. Second, [Lopez-Gomez . (2020)] sidesteps the
turbulence parameterization problem by using very highly resolved ($\Delta x$
= $\Delta y$ = 35 m and $\Delta z$ = 5 m) three dimensional LES, managing
computational expense by using a sparse ensemble as a library from which to
train eddy diffusivity / mass flux based parameterization schemes [Cohen .
(2020)]. Advantages of the highly resolved LES choice include a luxuriously
converged limit that sidesteps most need to parameterize beyond microphysics;
disadvantages include imposing idealizations of lateral periodicity and a
scale separation in their harness to a global host, as well as limited
geographic sampling due to the expense of such LES; however, the latter is
positioned to be managed with calibration schemes that may inform where such
calculations can be strategically deployed to maximum global benefit [Dunbar .
(2022)]. Advantages of the Eddy-Diffusivity Mass-Flux (EDMF) framework include
its interpretability and generalizability; disadvantages include its potential
inability to subsume some complicated organization feedbacks. Finally,
miyamoto2013deep avoids scale separations entirely by directly resolving fully
global uniform 870-m horizontal resolution. Similar work currently planned to
attain 200-m global horizontal resolution has advantages of resolving the
outer scale of boundary layer eddies without drawing scale separations, but
the disadvantage of the inordinate computational expense and inability to
conduct multidecadal cloud feedback experiments, as well as a limited ability
to refine vertical grids near the inversion. All of the above approaches must
cope with the ongoing difficulties and uncertainty of how to parameterize
microphysics.

We will focus on a strategy that is complementary to all the above approaches
for dealing with the computational challenge of low clouds for climate
simulation, by using the multiscale modeling framework (MMF, also referred to
as “superparameterization (SP)”;
grabowski2004improved,khairoutdinov2001cloud,hannah2020initial), in which a
coarse resolution ($\sim$100 km) global climate model (GCM) is coupled to an
embedded convection-resolving models (CRMs) at each global grid location. Many
previous studies [Kooperman . (20161), Kooperman . (20162)] have shown low-
resolution MMF (LR) tests with a traditional MMF, i.e. using coarse 4-km
horizontal resolution and greater than 100-m vertical spacing, can improve the
simulated rainfall distribution and wave spectrum near the equator and over
summer continents. However low-cloud-forming eddies are not directly resolved
in this approach, requiring subgrid scale parameterizations to cope with [Wang
. (2015), Cheng Xu (2015)].

The MMF coupling paradigm does not put constraints on the CRM domain size or
grid spacing, so large eddy permitting resolution can be used to sidestep most
of the parameterization problem regarding cloud-forming eddies. The MMF
approach makes the physical idealizations of imposing lateral periodicity in
the CRM and a scale separation between the GCM and CRM, and historically has
limited the CRM to just two dimensions for computational efficiency. Despite
these concessions the MMF has a unique computational advantage that enables
full geographic sampling and relatively fast throughput, with options to
accelerate the CRM algorithmically [Jones . (2015)] or via GPU hardware [WM.
Hannah . (2020)], and regionalized load balancing [Peng . (2022)] that makes
the MMF approach increasingly attractive for climate dynamics and low cloud
feedback applications.

A high-resolution MMF (HR) with a grid designed for low cloud simulations was
first explored by parishani2017toward with hopes of more directly simulating
shallow convection over Sc-covered regions. Encouraging initial improvements
in low cloud vertical structure, diurnal sensitivity, and the vertical
structure of sub-cloud turbulent kinetic energy were reported in a model
configuration using simplified bulk, one-moment cloud microphysics.

However, such HR experiments have to date been unable to sustain sufficient
liquid water in stratocumulus regions, where simulations suffer from an
undesired regional dim bias that has been difficult to overcome [Parishani .
(2017)]. Associated symptoms have implicated an unknown source of vertical
mixing that disrupts the balance required to sustain morning Sc by mixing too
much free tropospheric air into the boundary layer. The overall impact is to
under-predict daytime cloud liquid water resulting in too little time mean
shortwave reflectivity. Meanwhile, the assumptions inherent in an MMF that can
limit its ability to laterally advect condensed water between adjacent CRMs
have caused some to question its capacity to maintain low clouds [Jansson .
(2022)]. While the scale separation inherent in the MMF also introduces
distortions (such as the neglect of the mesoscale), it does allow a global
model to simulate these fine scales, making it possible to represent physical
processes (e.g., cloud top entrainment, aerosol activation in updrafts) that
drive critical sensitivities of low clouds to anthropogenic influence.

In short, the question is open as to whether the HR approach should ever be
expected to maintain realistic amounts of liquid in marine Sc regions, to the
extent that it must rely primarily on local cloud generation to succeed, and
given that over-entrainment has proved a stubborn problem to overcome.
Motivated by the Transpose-Atmospheric Model Intercomparison Project
[<]Transpose-AMIP, ¿williams2013transpose, we use the hindcast approach to
test different model configurations. In this context, the purpose of this
paper is to explore two mechanisms to control entrainment efficiency in a HR
and examine the extent to which they can alleviate the Sc dim bias issue. The
first is to numerically damp unphysical noise at the grid scale caused by the
numerics by applying a hyperdiffusive term (which we will refer to as
“hyperviscosity”) to the momentum equation that can reduce entrainment and
entrainment efficiency [Wyant . (2018)]. Second, enhancing cloud droplet
sedimentation (henceforth, “sedimentation”), which can also reduce the
entrainment efficiency and preserve cloud liquid by depleting the liquid water
in the cloud-top entrainment zone [Bretherton . (2007)]. We will use the term
”entrainment efficiency” in our study as an indicator of the magnitude of
tendencies resulting from the combined effects of explicitly resolved mixing,
numerical diffusion, and parameterized subgrid-scalediffusion.

The paper is organized as follows: Section 2 contains a rationale and
description of how we implement both hyperviscosity and sedimentation
processes. In Section 3, we first analyze the results of six, short-duration
sensitivity simulations in a testbed HR configuration to understand the impact
of varying degrees of CRM-scale hyperviscosity and sedimentation on low cloud
characteristics and the spectrum of turbulent eddies in the marine boundary
layer. These results point to temporal nonlinearities and a promising
configuration for longer-duration simulations in an operational configuration,
for which we show results from a subsequent round of microphysical tunings.
This allows an updated view of HR top of atmosphere radiative biases after
controlling for over-entrainment. A summary of the results and a discussion
are included in section 4.

##  2 Methods

###  2.1 Model Description

In this study, we use the Multi-scale Modeling Framework configuration of the
Energy Exascale Earth System Model (E3SM-MMF; hannah2020initial) as a testbed
to examine the impact of hyperviscosity and sedimentation on low clouds
simulated by high resolution embedded convection arrays. E3SM was forked from
the NCAR CESM [Hurrell . (2013)] but has undergone continued development and
enhancement since then [Golaz . (2019), Xie . (2018)]. The dynamical core uses
a spectral element method on a cubed-sphere geometry [Ronchi . (1996), Taylor
. (2007)]. Physics calculations are done on a finite volume grid that is
slightly coarser than the spectral element grid used for dynamics, but the
physics grid is comparable to the effective resolution of the dynamics grid
and does not alter the qualitative behavior of the model [WM. Hannah .
(2021)].

Each simulation follows the approach of [M. Khairoutdinov . (2005)] with a
two-dimensional CRM based on the System for Atmospheric Modeling [<]SAM,
¿khairoutdinov2005simulations embedded within each GCM physics column. These
embedded CRMs are oriented meridionally within the host GCM grid cell and have
periodic lateral boundary conditions. The vertical grid and background
anelastic state are updated to match the parent GCM column for each CRM
integration (typically once per GCM time step). The CRM uses a one-moment
microphysics scheme with a temperature-dependent partitioning of the cloud
condensate (cloud water and ice) and a subgrid-scale (SGS) turbulent transport
scheme with a diagnostic Smagorinsky closure. The rapid radiative transfer
model for General model application (RRTMGP) [Pincus . (2019)] is used for
radiation, which has been rewritten in C++ to run efficiently on GPUs. CRM
columns are combined for radiative calculations to reduce the computational
burden of radiation instead of considering each CRM column separately, which
does not qualitatively affect the model solution for typical configurations
with 4 or more radiative columns. Radiative tendencies are calculated once
each GCM time step and are applied back to the corresponding group of CRM
columns on the following time step. The domain average CRM variables for
temperature and water species are used to calculate forcing and feeedback
tendencies in order to couple the CRM and GCM, following the conventional MMF
coupling scheme described by grabowski2004improved.

###  2.2 Hyperviscosity

We now proceed to outline the first method envisioned to control MMF
stratocumulus entrainment in HR configurations. Models like SAM use
oscillatory centered difference numerical schemes [Wicker Skamarock (2002)]
for momentum advection and upstream biased discretizations of scalars with SGS
closures have relatively weaker performance than other numerical formulations
in an LES case study based on the first research flight (RF01) of the second
Dynamics and Chemistry of Marine Stratocumulus (DYCOMS-II) field campaign
[Pressel . (2017)]. wyant2018sensitivity find that in 3D LES simulations with
a horizontal resolution of 35 m and vertical grid spacing as fine as 5 m, a
hyperdiffusion term helps to increase LWP by numerically damping small-scale
eddies and reducing entrainment and entrainment efficiency. We test this
method in HR’s 2D CRM arrays by applying a hyperdiffusive term [Wyant .
(2018)] to the momentum equation inside the CRM.

The fourth-order hyperdiffusivity can be written as

$\partial_{t}{\vec{u}}=-k\nabla_{h}^{4}\vec{u},$ (1)

where $k=\Delta x^{4}/(16\,\tau)$ is the effective diffusivity and
$\nabla_{h}^{4}$ operator applies along the horizontal ($x$) direction of our
2D CRM using the fourth derivative central finite difference with second-order
accuracy. Compared to Laplacian diffusivity $\nabla^{2}$, hyperdiffusivity can
more selectively damp the smallest-scale structures confined in smaller
wavelength ranges, with little impact on larger scales [Maron . (2008)]. This
value of $k$ damps oscillations with Nyquist wavelength on a time scale of
$\tau$, which has a default value of 30 s in our simulations.

It is important to note that in LR configurations with $\Delta x=1200$ m and a
Nyquist wavelength of 2400 m, such hyperviscosity should be expected to be
counterproductive, given that LR low clouds rely on under-resolved grid-scale
“eddies” to deliver moisture from the surface; in this context, hyperviscosity
should be expected to shut down low cloud formation unhelpfully, something we
have confirmed (not shown). But in our HR configurations with $\Delta x=200$ m
the use of a filter like Equation 1 is better posed given that the cloud-
forming boundary layer eddies occupy multiple horizontal grid columns. Put
another way, only as MMFs have exited their infancy to allow sub-km horizontal
resolution, has CRM-scale hyperviscosity become an interesting consideration.

###  2.3 Sedimentation

Our second method to control entrainment efficiency considers a slight
modification of HR’s simple microphysics. In the one-moment microphysics
scheme used in our simulations, condensation occurs when the water vapor
amount exceeds saturation, with the excess above saturation converted to
liquid, ice, or a mixture of the two depending on temperature. While
precipitating liquid and ice (rain, snow, and graupel) sediment as described
in khairoutdinov2003cloud and heymsfield2003properties, cloud liquid droplets
did not sediment in this scheme. Such droplets do sediment in reality and the
inclusion of sedimentation in numerical models leads to liquid water retention
and increased in-cloud liquid water content [Bretherton . (2007)]. Here, we
follow yau1996short and define the precipitation flux of cloud liquid due to
sedimentation as

$P=c[3/(4\pi\rho_{l}N_{d})]^{2/3}(\rho q_{c})^{5/3}exp(5ln^{2}\sigma_{g}),$
(2)

where $\rho$ is the air density of air, $\rho_{l}$ the water density, $N_{d}$
the cloud droplet number concentration, $q_{c}$ the cloud liquid water mixing
ratio, $\sigma_{g}$ the geometric standard deviation of the (lognormal) cloud
droplet size distribution, and $c=1.19\times 10^{8}$ $m^{-1}s^{-1}$. The cloud
droplet number concentration is prescribed as a constant 70 $cm^{-3}$ over
ocean and 140 $cm^{-3}$ over land. A larger value of $\sigma_{g}$ corresponds
to a broader size distribution and a faster terminal velocity for larger size
droplets. geoffroy2010parametric provide estimates of $\sigma_{g}$ based on
two marine field campaigns, with a central estimate of 1.34 and a
parameterization of $sigma_{g}$ with values ranging from 1.2 to 1.5 for liquid
water contents ranging from 0.01 to 2 g m-3. Below, experiments with fixed
values of $\sigma_{g}=$ 1.2 and 1.5 are used to characterize the impact of
sedimentation in HR MMF.

The cloud optical depth is closely related to the cloud fraction, effective
cloud droplet radius, and the in-cloud liquid/ice water content. Following the
previous implementation of the single-moment microphysics, fixed values of
cloud effective radius for land and ocean are used for all simulations. Only
cloud fraction and liquid/ice water content affect cloud optical properties.
As we will see, incorporating these effects of sedimentation will reduce
entrainment efficiency by drawing liquid down from the inversion zone,
especially for larger values of $\sigma_{g}$.

###  2.4 Experimental Design

By default, E3SM-MMF uses a $60$ level vertical grid (L60), which is coarser
than the default $72$ level grid (L72) used by E3SM. The L72 grid was
implicated as the cause of intermittent numerical instability in E3SM-MMF due
to very thin layers near the surface, which is often around $20$ m thick in
the lowest layer. Instead of addressing this by reducing the time step the L60
grid was designed to avoid instability with an approximate thickness of $100$
m in the lowest level. Alternatively, the simulations presented here utilize a
$125$ level grid (L125) that is designed to concentrate refinement roughly
between $500$ and $1800$ m to improve the representation of sharp temperature
inversions needed to represent marine stratocumlus clouds. A smaller CRM time
step is used to avoid any numerical issues. Thus, the configuration referred
to as “LR” in this study is not the classical cloud SP that uses both coarse
vertical and horizontal resolution; rather it can be compared to the
“C32-L125-250m” MMF grid configuration of parishani2017toward. According to
[Bretherton . (1999)], this vertical grid spacing is not sufficient to resolve
entrainment but reflects a pragmatic choice that is computationally affordable
in the E3SM-MMF and is intentionally consistent with parishani2017toward.

In Sections 3.1 and 3.3, a computationally efficient configuration will be
exploited for hindcast experiments by using a relatively coarse ne16pg2 global
grid with 6,144 columns for physics calculations (approximately $2.8$ degree
grid spacing). These simulations use a $10$ min GCM physics time step. To
ensure computational efficiency, the radiation calculations were constrained
to $16$ columns. This means that radiative heating was computed based on time
and spatial averages, considering evenly grouped CRM columns within each GCM
column. Each simulation was run for $15$ days starting from an initial
condition derived from European Centre for Medium-Range Weather Forecasts
(ECMWF) Reanalysis (ERA5) atmospheric data [Hersbach . (2020)] and National
Oceanic and Atmospheric Administration (NOAA) sea surface temperature and sea
ice data. All hindcast simulations are initialized from 1 October 2008. In our
LR hindcast configuration, the embedded CRM has $32$ columns with a horizontal
grid spacing of $1200$ m ($38.4$ km extent) and a $5$ s CRM time step (see
Table LABEL:tab:exp_summary). Our HR hindcast configuration we use $64$
columns with a horizontal grid spacing of $200$ m ($12.8$ km extent) and a
$0.5$ s CRM time step.

Table 1: A summary of the simulations performed in this study

Simulation ID | levels | N |  $dx$ (m) | Extent (km) |  $dt$ (s) |  $\tau$ (s) | $\sigma_{g}$
---|---|---|---|---|---|---|---
LR | 125 | 32 | 1200 | 38.4 | 5 | - | -
HR | 125 | 64 | 200 | 12.8 | 0.5 | - | -
HRh | 125 | 64 | 200 | 12.8 | 0.5 | 30 | -
HRh15 | 125 | 64 | 200 | 12.8 | 0.5 | 15 | -
HRs15 | 125 | 64 | 200 | 12.8 | 0.5 | - | 1.5
HRhs12 | 125 | 64 | 200 | 12.8 | 0.5 | 30 | 1.2
HRhs15 | 125 | 64 | 200 | 12.8 | 0.5 | 30 | 1.5

$\bullet$ Note: $N$=number of CRM columns; $dx$=CRM horizontal resolution;
$\tau$=damping time scale; $dt$= CRM time step; $\sigma_{g}$=the logarithmic
width of the droplet size distribution, for the simulations that included
sedimentation effects.

The first two hindcast experiments are used to compare the LR and HR
configurations (first two rows of Table LABEL:tab:exp_summary). The rest of
the hindcasts are based on the HR configuration and perturb the magnitude of
$\tau$ in Eq.1 and $\sigma_{g}$ in Eq.2. In HRh, we add hyperdiffusion, with
$\tau=30$ s. In HRh15, we test the model sensitivity to the damping time scale
with $\tau=15$ s. Halving $\tau$ doubles the magnitude of $k$ in Eq.1, which
intensifies the damping of small-scale turbulent eddies; this will turn out to
have some encouraging but insufficient improvements in low cloud amount. The
HRhs12 and HRhs15 simulations combine hyperdiffusion (with $\tau$ fixed at 30
s) with perturbed cloud drop size distributions using $\sigma_{g}=1.2$ and
1.5, respectively. As we will see, it is these latter experiments that produce
the most encouraging improvement in the stratocumulus dim biases that have
hampered past incarnations of HRh.

In section 3.4, we explore more computationally abitious simulations using a
ne30pg2 global grid with 21,600 physics columns (approximately $1.5$ degree
grid spacing). These simulations are similar to the HR configuration described
above but with several notable differences in their configuration,
specifically a $20$ min GCM physics time step, a $2$ s CRM time step, $256$
CRM columns with a horizontal grid spacing of $200$ m ($51.2$ km domain
extent), and $4$ radiative columns. Another important difference of these runs
is that they utilize schemes for convective momentum transport [Tulich (2015),
Yang . (2022)] and CRM variance transport [W. Hannah Pressel (2022)], which
have recently been shown to improve various aspects of E3SM-MMF. Each tuning
experiment was run for six months, from January to June, using seasonally-
varying climatological conditions based on the years 2005–2014. This ambitious
ensemble was made possible by ongoing development to enhance the throughput of
E3SM-MMF, which includes code refactoring to leverage GPU hardware
acceleration on the Oak Ridge Leadership Computing Facility (OLCF) Summit
machine [Norman . (2019)].

Figure 1: Low cloud fraction differences based on the first day average (left)
and two-week averaged (right) between (ab) LR and HR (cd) HRh and HR, (ef)
HRh15 and HR, (gh) HRhs12 and HR, (ij) HRhs15 and HR, and (kl) HRs15 and HR

##  3 Results

###  3.1 General Features of the Simulation Based on Global Hindcast Results

Because the HR configuration has been identified as suffering from a deficit
of low cloud in stratocumulus regions (by as much as 20% in the Sc covered
ocean), we simulate two week hindcasts from October 2008 with the model
configurations in Table LABEL:tab:exp_summary and seek those configurations
that produce sustained more low cloud relative to HR. Figure 1 shows the
change in low cloud fraction relative to HR for each model configuration, with
results from both the first day and the second week of the simulations used to
identify the initial and longer-term responses. As in parishani2017toward, the
LR configuration produces increased cloud cover relative to HR, with
widespread increases that are not focused in the stratocumulus regions (Figure
1a,b). The first encouraging result is that selectively damping small size
eddies can retain more subtropical stratucumulus clouds during the first
simulated day than HR (geographic patterns in Figure 1c,e), but unfortunately
that improvement is transient so that this initial effect is not sustained
over a two-week average (Figure 1d,f). While enabling sedimentation of cloud
droplets provides modest ($\sim$0.1%) increases in low cloud in some
stratocumulus regions (Figure 1k,l), combining sedimentation with
hyperviscosity leads to an even stronger initial stratocumulus cloud increase
(Figure 1g,i) that is sustained on longer timescales (Figure 1h,j), suggesting
a viable path to improve HR’s climatological stratocumulus bias. Increasing
the droplet size broadness parameter amplifies this effect (Figure 1j).

If one estimated the low cloud change with the combined effects of
hyperviscosity and sedimentation in HRhs15 (Figure 1j) as the sum of their
individual impacts in HRh and HRs15 (Figure 1d,l), the estimate differs
greatly from the result in HRhs15 in both its magnitude and in the regional
distribution of cloud changes. This nonlinear response in HRhs15 concentrates
low cloud increases in the stratocumulus regions, suggesting the synergistic
interactions of hyperviscosity and cloud droplet sedimentation lead to more
persistent stratocumulus clouds. We believe that these clouds are sustained by
more realistic turbulent circulations within the marine boundary layer and
will explore this further in section 3.3.

Figure 2: Averaged absorbed shortwave radiation (ASR) differences between (a)
LR and HR, (b) CERES and HR, (c) HRh and HR, (d) HRh15 and HR, (e) HRhs12 and
HR, and (f) HRhs15 and HR.

We next examine whether these cloud fraction increases are radiatively
significant relative to the shortwave stratocumulus dim bias of concern. Based
on the two-week averages, Figure 2b first calibrates its structure and
magnitude relative to CERES data in our simulations: Note the collection of
negative ASR anomalies over the Sc regions makes a large contribution to the
RMSE. Unlike other regional details of the absorbed shortwave radiation (ASR)
biases that are difficult to disentangle from internal variability in a 15-day
sample, the consistent ASR bias over both of the subregions of most
subtropical Sc during October (off the western coasts of Namibia and Peru)
indicates a robust climatological signal. Despite using a different dynamical
core and modeling framework from parishani2017toward, this baseline ASR bias
is similar to their C32-L125-250m simulation (their Figure 4b) indicating some
stable signals across different MMF implementations.

We now look at change in ASR from this baseline for the simulations with
hyperviscosity and sedimentation (Figure 2c-f). Consistent with its effects on
cloud fraction, when hyperviscosity is used in isolation (both HRh and HRh15;
Figure 2c-d) there is no reduction in the dim bias over Sc regions; rather,
the dim bias becomes slightly more pronounced in those simulations. When both
effects are combined, cloud induced brightening occurs throughout the
stratocumulus regions (relative to HR), except in the near-coastal
environment, acting encouragingly in the same direction as the baseline
anomalies (negative ASR anomalies off the west coasts of the Namibia, Peru,
and Western Australia in Figure 2f). This reduced ASR dim bias corresponds
well with the locations of low cloud fraction increase (Figure 1hj). None of
the four model perturbations introduce notable outgoing longwave radiation
(OLR) differences compared to HR based on the 15-day means (Figure 10).

Figure 3: The map of (a) highlighted regions used in Figure 4 and (b) the
selected regions to construct height-time plots over Peruvian (gray), West
Australian (red), and Namibian (blue).  Figure 4: A comparison of (a) the
meridional mean absorbed shortwave radiation (ASR) (b) outgoing longwave
radiation (OLR), (c) low cloud fraction, and (d) cloud liquid water path (LWP)
differences between LR and HR, HRh and HR, HR15 and HR, HRhs12 and HR, HRhs15
and HR, and CERES and HR. The three Sc regions, namely the west coast of
Australia, Peru, and Namibia, are represented by the orange, gray, and blue
shadings from left to right, respectively.

We now hone in on a strategic subregion of the Southern Hemisphere subtropical
Sc latitudes (highlighted in Figure 3a). Inter-model differences of
meridionally and time average properties along this zonal transect are shown
in Figure 4. The strongest and most interesting changes relative to HR occur
when hyperviscosity and sedimentation are combined in HRhs15, producing
encouraging Sc brightening that is emphasized in the thick, dark red line. The
thick black line serves as a reference, representing the comparison between
CERES data and the baseline HR simulation. Closeness to this line - as begins
to occur in the eastern parts of the Peruvian (grey) and Namibian (blue)
regions, indicates a reduced bias. To orient, the shaded regions (Figure 4)
delineate the three Sc zonal subregions highlighted in Figure 3a: off the west
coasts of Australia (orange shading), Peru (gray shading), and Namibia (blue
shading); in these regions, HRhs15 produced time-mean cloud brightening
relative to HR on the order of 20-35 W/m2. The other panels show that in these
same regions HRhs15 also produce 5-13 $\%$ more low cloud (Figure 4c) and
0.03-0.05 $kg/m^{2}$ larger cloud liquid water paths (Figure 4d). When
hyperviscosity is used in isolation (HRh and HRh15), cloud brightening occurs
over the Australian Sc deck relative to HR but the dim bias is worsened over
the Peruvian and Namibian Sc decks; that is, the effect is not systematic
across Sc regimes. Likewise, HRh and HRh15 have either roughly no change or
reduced cloud amount relative to HR and liquid water path for the two regions
where the dim bias worsens.

In summary, we find robust reductions in the two-week-mean HR dim bias over
the Sc regions in HRhs15 which occur due to increased cloud amount and liquid
water path. Changes seem to be due mainly to the synergistic effects of
sedimentation and hyperviscosity, as their combined effect is much larger than
when either is used in isolation.

Figure 5: Height time evolution of vertical velocity variance
($\overline{w^{\prime}w^{\prime}}$, in units of $m^{2}/s^{2}$) in Peruvian
averaged over 15 days starting from October 1st 2008. (a) LR, (b) HR, (c) HRh,
(d) HRh15, (d) HRhs12, (f) HRhs15. The blue lines represents the total grid-
box liquid water path and cyan line represents HRhs15 for all panels as a
reference. Gray shaded time intervals represent nighttime. Black contours are
showing the $10\%$ (black dashed line) and $20\%$ (black solid line) cloud
fraction.

###  3.2 Analysis of Peruvian Stratocumulus Region

This section focuses on the Peruvian Sc region to examine further details of
the unsteady evolution of boundary layer vertical structure and the associated
changes in low clouds. This region, lying off the west coast of South America
over the ocean (gray area in Figure 3b), is one of the most persistent Sc
decks [Bretherton Wyant (1997)] and poorly simulated in models [Konsta .
(2022)].

Figure 5 shows the time-height evolution of CRM-scale vertical velocity
variance (shading), a good proxy of low-level turbulent mixing, revealing its
co-evolution with cloud fraction (black contours). Both quantities are
averaged over the Peruvian Sc region (Figure 3b), as is the time series of
liquid water path shown below each contour plot. The cyan line benchmarks the
liquid water path in HRhs15, the simulation that resulted in the most liquid.
Strong diurnal cycles are apparent in all simulations, with strong turbulent
mixing occurring during local nighttime, as expected [Hignett (1991)].
Compared to LR, which uses embedded CRMs that are larger and have a much
coarser horizontal resolution, the baseline HR (Figure 5b) produces a larger
magnitude of $\overline{w^{\prime}w^{\prime}}$ which also extends to a higher
altitude, at least during the first few simulated days. These signals and
differences between HR and LR are consistent with parishani2017toward,
including the inability of HR to sustain low clouds beyond day 4, consistent
with its dim bias. Interestingly, in HR some nontrivial
$\overline{w^{\prime}w^{\prime}}$ is found above the cloud layer, whereas in
LR, above-cloud $\overline{w^{\prime}w^{\prime}}$ is near-zero. The
$\overline{w^{\prime}w^{\prime}}$ vertical structure for HR simulation appears
to have two modes during the first two days of simulation, with one mode near
the surface and the other mode closer to the cloud layer, suggesting
decoupling. On the other hand, LR only shows one local maximum
$\overline{w^{\prime}w^{\prime}}$ in the sub-cloud layer with a much weaker
magnitude. This local maximum $\overline{w^{\prime}w^{\prime}}$ occurs near
the surface during daytime and halfway between the surface and cloud level
during the nighttime. Although LR does not suffer from a particularly strong
over-entrainment bias [Parishani . (2017)], the cloud layer is supported
rather unrealistically by a weak $\overline{w^{\prime}w^{\prime}}$ maximum
[Hignett (1991), Heinze . (2015), Mechem . (2012)].

HR has a much smaller cloud fraction and LWP than the other simulations
(Figure 5b vs. others), and several symptoms implicate too much entrainment as
a key cause. For instance, HR has a much warmer sub-cloud layer temperature
(Figure 11) and this is systematic across Sc regions (Figures 14 and 15). The
warming cannot readily be explained by a difference in surface fluxes, given
that HR’s surface heat fluxes are roughly identical (Figure 11) to the other
HR configurations, especially during the first five days. We view HR’s warm
sub-cloud layer as a symptom of over-entrainment of warm overlying free-
tropospheric air: As a result of enhanced turbulence through and above the
cloud layer in HR, upward water transport is unable to sustain the cloud
against entrainment-driven warming and drying. In summary, we suspect that
strong $\overline{w^{\prime}w^{\prime}}$ near cloud top leads to over-
entrainment, and that this is the main cause of the dim bias over Sc regions
in the baseline HR simulation.

We now analyze our attempts to reduce over-entrainment, beginning with
applying hyperviscosity in isolation. We expect this to directly reduce
$\overline{w^{\prime}w^{\prime}}$ associated with small eddies regardless of
whether they were associated with moist processes. Indeed, above the cloud
layer, adding the hyperviscosity term with $\tau=30,15$ s helps to reduce the
above-cloud magnitude of $\overline{w^{\prime}w^{\prime}}$ (Figure 5c,d)
compared with HR (Figure 5b). Encouragingly, the low cloud fraction also
increases throughout the simulation. But the additional low cloud is only
recovered at night, which explains why hyperviscosity alone is not able to
alleviate the shortwave dim bias. Reducing $\tau$ from 30 to 15 s (HRh15)
helps to further reduce $\overline{w^{\prime}w^{\prime}}$ (Figure 5d), but has
minimal effects on low cloud fraction beyond those of HRh; indeed this is why
we use $\tau=30s$ as our default value for the hyperviscosity term. As pointed
out earlier, despite being encouraging, the effects of hyperviscosity alone
are not enough to fully address the over-entrainment problem that causes HR to
be unable to sustain enough low cloud. One signature of turbulence driven by
healthy amounts of cloud top radiative cooling is an elevated peak in
$\overline{w^{\prime}w^{\prime}}$ away from the surface; note that this is too
weak in HRh.

We now examine the impact of additionally including cloud droplet
sedimentation (Figure 5f). Removing cloud water away from the cloud top via
sedimentation results in a larger $\overline{w^{\prime}w^{\prime}}$ and a
promising improvement in overall low cloud fraction. As suggested by
bretherton2007cloud, this is driven by a reduction of the entrainment
efficiency due to reduced liquid in the cloud-top entrainment zone.
Reassuringly, the strongest $\overline{w^{\prime}w^{\prime}}$ is now found
well above the surface and in the upper half of the boundary layer, consistent
with cloud-top buoyancy production being the primary driver of convection,
especially during the nighttime as observed. Thick clouds persist beyond the
first week of the simulation as a result of the reduced consumption of cloud
liquid by entrainment. Again, these effects only occur in conjunction with
hyperviscosity. Incorporating sedimentation on its own results in a reduced
cloud fraction and a weaker $\overline{w^{\prime}w^{\prime}}$ (Figure 5e).

Table 2: Median values from day 2 to day 15 for selected variables during
daytime with nighttime values in parenthesis. This calculation discards $10\%$
of the upper and lower outlier points before estimating the median values.

| LR | HR | HRh | HRhs15
---|---|---|---|---
Cloud fraction ($\%$) | 44 (73) | 27 (46) | 31 (63) | 49 (73)
$LWP(g/m^{2})$ | 19 (43) | 7 (19) | 10 (13) | 24 (55)
$z_{i}$ (m) | 1301 (1216) | 1414 (1358) | 1387 (1298) | 1266 (1212)
Cloud top (m) | 1212 (1157) | 1218 (1283) | 1215 (1224) | 1195 (1195)
Cloud base (m) | 732 (661) | 752 (705) | 753 (685) | 653 (618)
$z_{LCL}$ (m) | 630(591) | 649 (624) | 639 (602) | 560 (548)
$\alpha_{q}$ | 0.24 (0.22) | 0.22 (0.27) | 0.22 (0.26) | 0.23 (0.24)
$w_{e}$ (mm/s) | 3.6 (3.9) | 4.3 (3.8) | 4.3 (3.8) | 4.0 (3.8)
$SWCRE(W/s^{2}$) | 59 (-) | 23 (-) | 32 (-) | 71 (-)

$\bullet$ Note: $LWP$=Total grid-box cloud liquid water path;
$z_{LCL}$=Lifting condensation level; $z_{i}$=Inversion height;
$\alpha_{\theta}$=decoupling parameters for potential temperature;
$\alpha_{q}$=Decoupling parameters for potential water vapor;
$w_{e}$=Entrainment rate (estimated as the subsidence rate at z=zi, assuming a
steady state); $SWCRE$=Shortwave cloud forcing (values are all negative during
the daytime).

Differences in inversion height are subtle to detect visually, but Table
LABEL:tab:AllCaseAvgValue summarizes median properties from cloudy grid points
(nonzero liquid water) in each simulation (averaged between days 2 to 15). The
inversion height ($z_{i}$) shown in Table LABEL:tab:AllCaseAvgValue generally
agrees with what was observed during the Ocean–Cloud–Atmosphere–Land Study
Regional Experiment (VOCALS-REx) at Point Alpha in October and November 2008
off the coast of South America, which varied between 996 m to 1450 m [Dodson
Small Griswold (2021)]. We note that HRhs15 has the smallest difference
between the cloud top height and the inversion height while HR has the
largest, which we connect to inversion strength in the next section.

Table LABEL:tab:AllCaseAvgValue also quantifies the decoupling that was
discussed subjectively in Figure 5. We estimated the decoupling parameters for
potential water vapor ($q$) as
$\alpha_{q}=\frac{q_{cld}-q_{ml}}{q_{inv}-q_{ml}}$ [Park . (2004)]. Subscripts
$cld$, $inv$, and $ml$ refer to mean values between the cloud base and top,
between the surface and cloud base, and at the inversion height respectively.
A decoupling parameter close to zero indicates a well-mixed boundary layer.
Previous observations suggest that the boundary layer is decoupled when the
parameter exceeds about 0.30 [Albrecht . (1995)]. HR produces the highest
values for $\alpha_{q}$ during the nighttime. There is much less contrast in
daytime than nocturnal decoupling in HRhs15 compared with other
configurations. A vertically decoupled thermodynamic structure produces cloud
bases well above the LCL [Miller . (1998)]; indeed the cloud base is 103 $m$
(81 $m$) above the LCL for HR during the daytime (nighttime), while 93 $m$ (70
$m$) for HRhs15. In the baseline HR, more decoupling can be viewed as a
symptom of over-entrainment that is likely to cause less Sc during the daytime
in that reduced moisture supply at cloud base cannot overcome the dry air
entrainment from cloud top. HRhs15 corresponds to a lower entrainment rate
($w_{e}$) during the daytime than HR and HRh. HRhs15 also corresponds to the
largest magnitude of shortwave cloud effects (SWCRE) due to a larger cloud
fraction.

Figure 6: Averaged vertical profiles nomalized by $z_{i}$ from day 2 to day 15
for (a) T total physics tendency ($K/s$), (b) cloud liquid water content
(g/kg), and (c) $\overline{w^{\prime}w^{\prime}}$ ($m^{2}/s^{2}$). Solid lines
represent night time average, while dashed lines represent the day time
average.

###  3.3 Composite vertical structure and turbulent scale analysis

We now examine mean daytime (dashed) and nighttime (solid) vertical profiles
(Figure 6) from days 2 to 15 over the Peruvian region. Here, the height, $z$,
is normalized by the inversion height ($z_{i}$) to give a nondimensional
vertical coordinate, $z/z_{i}$.

All configurations show a large diurnal cycle: the daytime cloud liquid water
content is about half of its nighttime value. HRhs15 (HR) corresponds to the
largest (lowest) cloud liquid water content in both nighttime and daytime
groups. It is interesting to note that HRhs15 has an even higher daytime cloud
liquid water content than nighttime HR; in fact, in the following section we
will show that there is too much daytime liquid in HRhs15, motivating
compensatory microphysical retuning.

Longwave radiative cooling at the cloud top is regarded as the primary driver
of convection in stratocumulus clouds [Lilly (1968), Nicholls (1989), Moeng .
(1996)]. Note that the peak cloud top radiative cooling that has the largest
contribution to the total temperature tendency (Figure 6a) occurs at the cloud
top [van Zanten (2002)]. Regardless of the large magnitude of
$\overline{w^{\prime}w^{\prime}}$ (Figure 6c), HRhs15 has slightly lower cloud
top radiative cooling compared with LR and HRh (Figure 6a) due to less liquid
emissivity near the cloud top. The level corresponding to the highest cloud
liquid water content (Figure 6b) is similar for LR, HR, and HRh, but this
level is lower and further away from the cloud top for HRhs15 due to
sedimentation. While HRhs15 and HR have a similar magnitude of cloud top
radiative cooling, this radiative cooling produces a thicker cloud and drives
stronger and better coupled vertical motions $\overline{w^{\prime}w^{\prime}}$
in HRhs15 than HR due to its weaker entrainment. A smaller magnitude of cloud
top radiative cooling for HR during the daytime might also help to explain the
warmer sub-cloud layer temperature compared with HRh, especially after day 7
shown in Figure 11a. On the other hand, this warming is less severe compared
between HR and HRhs15 near the cloud top.

Figure 7: A comparison of vertical profiles of the spectral intensity for
nighttime average of (a) LR, (b) HR, (c) HRh, and (d) HRhs15, and daytime
average of (e) LR, (f) HR, (g) HRh, and (h) HRhs15.

It is still not clear whether cloud top entrainment is controlled by small
eddies or large eddies [Wood (2012)]. To more closely examine the eddy spectra
in cloudy regions, we perform spectral analysis along the CRM’s horizontal
dimension looking at the power spectrum of the CRM-scale vertical velocity,
separately for daytime and nighttime (Figure 7).

Compared with HR (Figure 7bf), the spectral intensity distribution for HRhs15
(rightmost column) is clearly confined within the stable boundary layer (STBL)
with a single peak near 0.5$z_{i}$.The magnitude of this peak is larger during
the nighttime since nighttime has a larger LWP. In HRh, the spectral intensity
distribution peaks at towards larger wavelengths (Figure 7cg). In HRh, the
spectral intensity of small eddies (wavelengths close to $2\Delta x=400$ $m$)
is reduced, while the spectral power above the cloud top is enhanced with the
large eddies (wavelengths greater than 2000 $m$). HRh has the strongest
signals close to the largest wavelength that can be resolved (half of the
domain size). LR is able to resolve eddies with horizontal wavelengths of up
to 10 km thanks to its larger domain size (Figure 7ae). However, the
occurrence of the peak eddy spectral density for the smallest resolvable
eddies (2.4 km) indicates eddy variance pile-up on the grid-scale. This
implies that cloud-forming eddies, which are constrained by the numerical
grid’s finest resolved scale, are under-resolved in the low resolution
simulation. The HR configurations are more physically plausible
representations of sub-cloud turbulence in that a spectral peak exists
interior to the resolvable scales. The high resolution of HR permits sub-cloud
eddies to occupy multiple horizontal grid columns. We still can see relatively
strong eddies (wavelengths around 3000 $m$) above the cloud. While HRhs15 may
also cut off signals due to limited domain size, its spectral intensity
distribution clearly shows a much better range of the resolved signal and STBL
structure that is more consistent with observations and expectations from LES.

HRhs15 also corresponds to the weakest above cloud eddies to reduce
entrainment. Close to the inversion height ($z_{i}$), a stronger
stratification tends to reduce the magnitude of $E_{w}$ and the dominant
wavelength corresponding to the maximum $E_{w}$. Unlike HRhs15, relatively
large vertical velocity fluctuations above the inversion height in HR and HRh
corresponds well with a reduce the temperature inversion and, therefore, does
not have a significant impact on the wavelength near the cloud top (e.g.
Figure 7b). Above $z_{i}$, the eddies are much larger than they are below
$z_{i}$.

Figure 8: Absorbed shortwave radiation at TOA biases with respect to CERES for
the (a) HR, (b) HRhs15 with default configuration, (c) HRhs15 with
$q_{w}=5\times 10^{-4}$,(d) HRhs15 with $q_{w}=1\times 10^{-4}$, (e) HRhs15
with $q_{w}=5\times 10^{-4}$ and $q_{i}=8\times 10^{-5}$ (d) HRhs15 with
$q_{w}=5\times 10^{-4}$ and $q_{w}=5\times 10^{-5}$

###  3.4 Microphysics Tuning

The previous sections have provided evidence that the addition of
hyperviscosity and cloud droplet sedimentation produces encouraging changes in
stratocumulus clouds such as more daytime clouds. However, the coarse global
mesh, smaller CRM domains and short, two week duration of the hindcast
simulations analyzed above represent notable compromises when compared to
standard configurations of E3SM and E3SM-MMF. To address this concern we have
conducted a series of 6-month simulations using the ne30pg2 grid and a larger
CRM domain (Section 2.4) that is more typical for E3SM experiments. These
simulations are a subset of a larger tuning effort that considered several
microphysical parameters. Ultimately, we found that autoconversion thresholds
for liquid ($q_{cw0}=1\times 10^{-3}$ by default) and ice ($q_{ci0}=1\times
10^{-4}$ by default) were the most effective parameters for bringing the TOA
energy fluxes into a reasonable balance and so that is what we will focus on
below. This tuning exercise was partially motivated by the observation that
the low-cloud enhancement resulting from the use of hyperviscosity and cloud
droplet sedimentation produced a dramatic change in the TOA net shortwave
radiative flux (Figure 8b).

Table 3: The bias and RMSE in parenthesis for ocean and land | $q_{cw0}$ | $q_{ci0}$ | ASR (W m-2) | OLR (W m-2)
---|---|---|---|---
(kg kg-1) | (kg kg-1) | ocean | land | ocean | land
HR | 1$\times$10-3 | 1$\times$10-4 | 4.0(9.5) | 1.5(4.4) | -8.2(8.6) | -2.0(2.8)
HRhs15 | 10-3 | 1$\times$10-4 | -19.5(18.6) | -2.1(4.3) | -7.7(8.1) | 1.5(3.2)
HRhs15 | 5$\times$10-4 | 1$\times$10-4 | -10.4(11.3) | -1.1(4.4) | -5.9(7.0) | 0.1(3.2)
HRhs15 | 1$\times$10-4 | 1$\times$10-4 | 3.0(9.7) | 2.6(4.4) | -3.9(6.2) | -1.1(2.7)
HRhs15 | 5$\times$10-4 | 8$\times$10-4 | -8.6(10.3) | -0.2(4.1) | -4.8(6.1) | 1.3(3.1)
HRhs15 | 5$\times$10-4 | 5$\times$10-4 | -6.9(8.9) | 1.8(4.1) | -2.8(4.7) | 2.9(2.8)

Figure 8 shows a comparison of the absorbed shortwave radiation climatological
biases from our ten-year simulation compared with satellite observations. The
baseline HR and several retunings of the baseline HRhs15 configuration are
compared. Unlike HR, the ASR bias for HRhs15 primarily stems from too bright
marine clouds, especially over the subtropical Sc regions (Figure 8).
Especially strong negative ASR biases are found off the western coasts of
Peru, Namibia, Australia, and California. Reducing the liquid autoconversion
thresholds increases the ASR (Figure 8 bcd), approximately halving the global
mean shortwave bias, and reduces the RMSE. A reduced liquid autoconversion
threshold combined with an increased ice autoconversion threshold further
helps to ameliorate ASR global mean bias and reduce RMSE (Figure 8ef). The
configuration with $q_{cw0}=5\times 10^{-4}$ and $q_{ci0}=5\times 10^{-5}$
produces the smallest global mean bias and RMSE (Figure 8e). Most of this
improvement occurs over the ocean (Table LABEL:tab:SROLR_ocnlnd). Even with
the microphysics retuning, the incorporation of hyperviscosity and
sedimentation continues to play a crucial role in reducing the ASR bias
particularly over regions covered by stratocumulus clouds (Figure 16).

Figure 9: Outgoing longwave radiation at TOA biases with respect to CERES for
the (a) HR, (b) HRhs15 with default configuration, (c) HRhs15 with
$q_{w}=5\times 10^{-4}$,(d) HRhs15 with $q_{w}=1\times 10^{-4}$, (e) HRhs15
with $q_{w}=5\times 10^{-4}$ and $q_{i}=8\times 10^{-5}$ (d) HRhs15 with
$q_{w}=5\times 10^{-4}$ and $q_{w}=5\times 10^{-5}$

Microphysical tuning results in weaker changes of OLR than those for ASR
(Figure 9). Sedimentation (HRhs15) only slightly increased the global mean OLR
(Figure 9ab) from the base HR simulation, which was too opaque in the tropics.
This bias is reduced in Figure 9f. Overall, we are able to obtain less than
$1W/m^{2}$ OLR bias.

##  4 Discussion and Conclusions

Compared to other available global modeling tools for studying cloud feedback,
today’s GPU-accelerated Multiscale Modeling Frameworks configured with High
Resolution (HR) interior grids have the capacity to provide a unique
combination of global eddy-permitting resolution coverage and multi-decadal
throughput that complements other climate simulation technology. In theory HR
MMFs should be attractive for low cloud feedback analysis and cloud-aerosol
interactions, by making minimal assumptions about the sub-km scale vertical
eddy field.

But in practice, this depends on the model’s ability to represent present-day
climate. For over five years since the first experiments with HR MMF, it has
been unclear whether a chronic over-entrainment bias preventing realistic
amounts of Sc liquid water was surmountable. It has been natural to wonder if
the inherent idealizations of MMFs that make them computationally attractive –
i.e. the limited domain size, dimensionality, moderate (200-m) interior
horizontal resolution, lateral periodicity, and associated inability to
laterally advect liquid water conservatively – [Muller Held (2012), Jansson .
(2022)] – might impose fundamental limitations.

Our results argue otherwise: We suggest MMFs are simply in their infancy and
their interior resolved scale has never been sufficiently tuned to succeed in
a HR limit. To show this, we investigated the impact of adding hyperviscosity
and sedimentation on low cloud formation in a high resolution multi-scale
modeling framework (HR) that uses 200-m horizontal, and as fine as 20-m
vertical, grid spacing within each of its embedded convection resolving
models, configured with bulk one-moment microphysics. As in previous studies,
our control HR simulation produced the familiar bias of too few low clouds
over regions of subtropical marine stratocumulus (Sc), resulting in a dim bias
compared with satellite observations of shortwave radiation absorbed at top of
atmosphere.

We found promising Sc-selective brightening when we combined scale selective
damping (hyperviscosity) of grid scale momentum variations with the
introduction of cloud droplet sedimentation. The application of hyperviscosity
alone, which directly reduces $\overline{w^{\prime}w^{\prime}}$, leads to
short-lived increases in nocturnal cloud thickness. Simulations with
sedimentation alone lead to modest increases in cloud fraction. However, the
most encouraging effects occur when the two are applied together, whereupon
robust increases in cloud liquid water lead to a reversal of the Sc dim bias,
including in multi-week integrations. These nonlinear interactions of these
two processes lead to much stronger changes than when they are applied
separately. The resulting, larger peak liquid water concentration is shifted
downward, away from the cloud-top entrainment zone. In this configuration,
dense, locally-formed Sc are observed to form in the HR MMF, and the sub-cloud
eddy spectrum becomes especially well organized.

In summary, with only these minor, physically-motivated re-tunings, the CRMs
of a HR MMF can be coerced to create healthy amounts of locally generated
stratocumulus liquid, in association with reasonable sub-cloud eddy
properties. This is possible despite the assumptions of periodicity,
dimensionality (2D) and only 200-m horizontal resolution that makes HRs
computationally efficient, which is encouraging. At first, the interventions
create too much low cloud, and swap a regional dim bias for a global ocean
bright bias – but with encouragingly little horizontal variation across the
oceanic cloud regimes, with hope for calibration. Thus, as must occur
following any manipulation of a MMF’s physical formulation, we performed a
compensatory microphysical re-tuning to recover a reasonable top of atmosphere
climatology. Despite a limited tuning campaign, the results demonstrate the
potential for significantly less severe Sc dim biases, and reduced spatial
RMSE of shortwave absorbed radiation across the global ocean. It is logical to
expect that with further attention to tuning, even more operationally
attractive configurations could be uncovered.

Several limitations of this work are worth mentioning. In our current study,
we did not fully explore the contribution of resolved scale advection and
subgrid scale diffusion to the reduced entrainment efficiency caused by
hyperviscosity. With enhanced outputs including the SGS contribution in SAM,
future investigations could further evaluate these aspects to gain a deeper
understanding of the underlying mechanisms. We speculate a root problem
motivating the need to apply hyperviscosity in our simulations may be the
numerics of the momentum solver in the embedded CRM (See Section 2.1).
Successors to SAM under development by DOE for use in E3SM-MMF, like most
modern LES [Eldred (2021)], intentionally use a cell-centered, entropy stable
Weighted Essentially Non-Oscillatory (WENO) schemes for the momentum solver,
as suggested in the work of [Pressel . (2017)]. Another obvious limitation is
that the HR here uses a simple one-moment microphysics scheme and diagnostic
turbulence scheme. While this is helpful for maximizing throughput at its
ambitious grid resolution, it is also outdated. The eventual higher-order
microphysics [Morrison Milbrandt (2015)] that are expected to come online in
the E3SM-MMF may suffer less from baseline over-entrainment due to already
including a representation of the sedimentation process that we have argued
helpfully draws cloud liquid down from the inversion to optimize entrainment
efficiency. Perhaps this imminent next generation of HR will have less need
for compensatory CRM-scale tuning and less sensitivity to grid spacing to
achieve its low cloud potential. In our current study, we did not fully
explore the contribution of resolved scale advection and subgrid scale
diffusion to the reduced entrainment efficiency caused by hyperviscosity. With
the enhanced outputs of the SGS contribution in SAM, future investigations
could further evaluate these aspects to gain a deeper understanding of the
underlying mechanisms. Several limitations of this work are worth mentioning.
In our current study, we did not fully explore the contribution of resolved
scale advection and subgrid scale diffusion to the reduced entrainment
efficiency caused by hyperviscosity. With enhanced outputs including the SGS
contribution in SAM, future investigations could further evaluate these
aspects to gain a deeper understanding of the underlying mechanisms. We
speculate a root problem motivating the need to apply hyperviscosity in our
simulations may be the numerics of the momentum solver in the embedded CRM
(See Section 2.1). Successors to SAM under development by DOE for use in E3SM-
MMF, like most modern LES [Eldred (2021)], intentionally use a cell-centered,
entropy stable Weighted Essentially Non-Oscillatory (WENO) schemes for the
momentum solver, as suggested in the work of [Pressel . (2017)]. Another
obvious limitation is that the HR here uses a simple one-moment microphysics
scheme and diagnostic turbulence scheme. While this is helpful for maximizing
throughput at its ambitious grid resolution, it is also outdated. The eventual
higher-order microphysics [Morrison Milbrandt (2015)] that are expected to
come online in the E3SM-MMF may suffer less from baseline over-entrainment due
to already including a representation of the sedimentation process that we
have argued helpfully draws cloud liquid down from the inversion to optimize
entrainment efficiency. Perhaps this imminent next generation of HR will have
less need for compensatory CRM-scale tuning and less sensitivity to grid
spacing to achieve its low cloud potential. In our current study, we did not
fully explore the contribution of resolved scale advection and subgrid scale
diffusion to the reduced entrainment efficiency caused by hyperviscosity. With
the enhanced outputs of the SGS contribution in SAM, future investigations
could further evaluate these aspects to gain a deeper understanding of the
underlying mechanisms.

Then again, perhaps not. For now, it is clear that “multi-scale” modeling
frameworks seem to merit careful “multi-scale” physics calibration, and that
this has largely been overlooked on the interior resolved scale, at least in
explorations of MMF at the limit of HRs’ grid resolutions. On the one hand,
this annoyingly complicates the art of global model tuning. On the other hand,
it is good news for the long term potential to study low-cloud feedbacks
quasi-explicitly via the HR MMF approach. Despite its idealizations, healthy
amounts of present-day Sc cloud can evidently be recovered in an HR MMF,
allowing its computational advantages to be brought to bear on questions of
cloud feedback. It will be important to determine whether this modifies
previous estimates of the HR MMF low cloud feedback to warming [Parishani .
(2018)] from previous generation simulations that have struggled to capture
sufficient baseline low cloud.

## Open Research

All E3SM source code can be accessed from the GitHub repository of the
Project_Energy_Exascale_Earth. All code modifications needed to implement our
approach within a legacy fork of the E3SM MMF climate model can be found in
the repository of Peng_E3SM_with_hyperviscosity. The raw output data is
archived and can be accessed from Zenodo under the references of [Peng .
(20231)] and [Peng . (20232)].

###### Acknowledgements.

L. Peng, M. Pritchard, P. N. Blossey, and C. S. Bretherton acknowledge funding
from the National Science Foundation (NSF) Climate and Large-scale Dynamics
program under grants AGS-1912134 and AGS-1912130. M. Pritchard, L. Peng, and
A. M. Jenney further acknowledge DOE BER (DE-SC0023368). The authors
acknowledge the Texas Advanced Computing Center (TACC) at The University of
Texas at Austin for providing HPC, visualization, database, or grid resources
that have contributed to the research results reported within this paper. URL:
http://www.tacc.utexas.edu This research was supported by the Exascale
Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department
of Energy Office of Science and the National Nuclear Security Administration
and by the Energy Exascale Earth System Model (E3SM) project, funded by the
U.S. Department of Energy, Office of Science, Office of Biological and
Environmental Research. This work was performed under the auspices of the U.S.
Department of Energy by Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344. This research used resources of the Oak Ridge Leadership
Computing Facility, which is a DOE Office of Science User Facility supported
under Contract DE-AC05-00OR22725.

##  Appendix A Figures

Figure 10: Similar to Figure 2 but for outgoing longwave radiation (OLR)
difference.  Figure 11: Height time evolution of temperature difference over
Peruvian between HR and (a) HRh, and (b) HRh15, (c) HRhs12, and (d) HRhs15.
Surface sensible (dashed lines) and latent heat (solid lines) flux are shown
by blue lines for HR and green lines for other configurations. Two cloud
fraction contour lines for other configurations been subtracted by HR are
shown for 0.1 (black dotted line) and 0.2 (black solid line). Figure 12:
Height time evolution of vertical velocity variance
($\overline{w^{\prime}w^{\prime}}$, in units of $m^{2}/s^{2}$) in West
Australia averaged over 15 days starting from October 1st 2008. (a) LR, (b)
HR, (c) HRh, (d) HRh15, (d) HRhs12, (f) HRhs15. The blue lines represents the
total grid-box liquid water path and thick cyan line represents HRhs15 for all
panels as a reference. Gray shaded time intervals represent nighttime. Black
contours are showing the $10\%$ (black dashed line) and $20\%$ (black solid
line) cloud fraction.  Figure 13: Height time evolution of vertical velocity
variance ($\overline{w^{\prime}w^{\prime}}$, in units of $m^{2}/s^{2}$) in
Namibian averaged over 15 days starting from October 1st 2008. (a) LR, (b) HR,
(c) HRh, (d) HRh15, (d) HRhs12, (f) HRhs15. The blue lines represents the
total grid-box liquid water path and thick cyan line represents HRhs15 for all
panels as a reference. Gray shaded time intervals represent nighttime. Black
contours are showing the $10\%$ (black dashed line) and $20\%$ (black solid
line) cloud fraction. Figure 14: Height time evolution of temperature
difference in west coast Australia between HR and (a) HRh, and (b) HRh15, (c)
HRhs12, and (d) HRhs15. Surface sensible and latent heat flux are shown by
blue solid and dashed lines for HR (blue) and other configurations (green).
Two cloud fraction contour lines are 0.1 (black dotted line) and 0.2 (black
solid line). Figure 15: Height time evolution of temperature difference in
Namibian between HR and (a) HRh, and (b) HRh15, (c) HRhs12, and (d) HRhs15.
Surface sensible and latent heat flux are shown by blue solid and dashed lines
for HR (blue) and other configurations (green). Two cloud fraction contour
lines are 0.1 (black dotted line) and 0.2 (black solid line). Figure 16: The
ASR bias differences between HR and HRhs15 (a) with microphysics tunning and
(b) without microphysics tunning.

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