resolution-revised.tex

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\begin{document}

\runningheads{A.~R.~Herrington and K.~A.~Reed}{CAM resolution sensitivity}

\title{On resolution sensitivity in the Community Atmosphere Model}

\author{Adam R. Herrington\corrauth \& Kevin A. Reed}
\address{School of Marine and Atmospheric Sciences, Stony Brook University, Stony Brook, NY 11794}

\corraddr{\url{adam.herrington@stonybrook.edu}}

\begin{abstract}
Advances in high performance computing make it possible to run atmospheric general circulation models (AGCMs) over an increasingly wider range of grid resolutions, using either globally uniform or variable-resolution grids. In principle, this is an exciting opportunity to resolve atmospheric process and scales in a global model and in unprecedented detail, but in practice this grid flexibility is incompatible with the non- or weakly- converging solutions with increasing horizontal resolution that have long characterized AGCMs. In the the Community Atmosphere Model (CAM), there are robust sensitivities to horizontal resolution that have persisted since the model was first introduced over thirty years ago; the atmosphere progressively dries and becomes less cloudy with resolution, and parameterized deep convective precipitation decreases at the expense of stratiform precipitation. This study documents a convergence experiment using CAM, version 6, and argue that a unifying cause, the sensitivity of resolved dynamical modes to native grid resolution feedback onto other model components and explain these robust sensitivities to resolution. The increasing magnitude of resolved vertical velocities with resolution are shown to fit an analytic scaling derived for the equations of motion at hydrostatic scales. This trend in vertical velocities results in an increase in resolved upward moisture fluxes at cloud base, balanced by an increase in stratiform precipitation rates with resolution. Compensating, greater magnitude subsiding motion with resolution has previously been shown to dry out the atmosphere and reduce cloud cover. Here it shown that {\color{red}{both the increase in condensational heating from stratiform cloud formation and greater subsidence drying contribute to an increase in atmospheric stability with resolution,}} reducing the activity of parameterized convection. The impact of changing vertical velocity field with native grid resolution cannot be ignored in any effort to recover convergent solutions in AGCMs, and in particular, the development of scale-aware physical parameterizations.
\end{abstract}

\keywords{Climate models, physical parameterizations, resolution sensitivity}

\maketitle

\section{Introduction}

An increasing number of Atmospheric General Circulation Models (AGCMs) are being developed to maximize efficiency on massively parallel systems, permitting regionally-refined high-resolution, or even globally high-resolution weather ($\Delta x \leq 5$ km) and climate ($\Delta x \leq 50$ km) simulations \citep{SMTMN2008JCP,MPASatm,Z2014QJRMS,HETAL2016JCLIM,DCMIP16,LetAl2018JAMES}. These models are built using unstructured meshes that {\color{red}{allow}} for substantial grid flexibility, but lack physical parameterizations ({\em{physics}}) which behave consistently as the truncation scale of the model changes with different grid resolutions, referred to as scale-aware physics. The most common approach towards developing scale-aware physics is through the lens of limited area, large-eddy simulations \citep[e.g.,][]{PC2008JAS,AW2013JAS,SZ2018JCLIM}. By subsequently filtering large-eddy solutions to lower-resolution grids, a relationship between first- and higher-order moments may be understood and ultimately parameterized as a function of grid resolution. While this approach is necessary for developing scale-aware physics, it is not sufficient. The equations of motions have inherent scale dependencies, and the properties of dynamical modes are a function of native grid resolution \citep{O1981JAS,WETAL1997MWR,PG2006JAS,JR2016QJRMS}. Scale-aware physics should also recognize these native grid dependencies.

{\color{red}{Conventional physics packages interact with the adiabatic dynamics to produce non- or weakly- converging solutions with increasing resolution \citep[{\em{resolution}} refers to {\em{horizontal resolution}} hereafter;][]{W2008TELLUS}. Although some characteristics of resolution sensitivity are model dependent, others are common across different AGCMs and indicate a common mechanism. For example, increasing resolution lends itself to greater stratiform precipitation and reduced parameterized convective precipitation in the Hadley Centre climate model \citep[HadAM3;][]{PS2002CD}, the Max Planck Institute for Meteorology model \citep[ECHAM5;][]{HETAL2006JC} and the National Center for Atmospheric Research (NCAR) Community Climate Model \citep[CCM2;][see Figure~\ref{fig:cam-history}]{WETAL1995CD}. So while the purpose of this study is to provide a process-level understanding of resolution sensitivity in community developed NCAR models, these results are useful for understanding resolution sensitivity in models that exhibit similar behavior.}}

The sensitivity of the Community Atmosphere Model \citep[CAM;][]{CAM5} and its predecessor CCM, to resolution is well documented through convergence studies \citep{KW1991JGR,WETAL1995CD,W2008TELLUS,RETAL2013JCLIM,ZetAl2014JCb,HR2017JCLIM}. CAM/CCM is a fully supported, well-funded climate model, but despite thirty years of continual model development, there are robust sensitivities to resolution that have persisted in all versions of the model. This study argues that a unifying cause, the inherent scale sensitivities of the underlying dynamical equations, can explain these robust responses to resolution that occur in CAM/CCM.

In CAM/CCM, the atmosphere progressively dries with increasing resolution, seen through a reduction in simulated total precipitable water \citep{KW1991JGR,WETAL1995CD,W2008TELLUS,RETAL2013JCLIM,ZetAl2014JCb,HR2017JCLIM}, which typically, but not always \citep[see][]{WETAL1995CD,ZetAl2014JCb}, coincides with a reduction in cloud cover. \cite{KW1991JGR} and \cite{WETAL1995CD} suggested that the drying of the atmosphere is due to greater magnitude resolved vertical velocities with increasing resolution, with greater subsiding motion increasing the export of dry air from the upper troposphere. This mechanism is consistent with an analysis of moisture budgets in CAM, version 4 \citep[CAM4;][]{CAM4} across multiple resolutions \citep{HR2017JCLIM}.

It is well known that the magnitude of vertical velocities increase with resolution in atmospheric models. While the cause of this sensitivity has been established for large-eddy simulations \citep[see][and references therein]{J2017JAMES}, only recently has the vertical velocity field in AGCMs and their sensitivity to resolution received attention \citep{DETALA2016ACP,OETAL2016JAMES}, albeit with seemingly conflicting explanations \citep{RETAL2016CD,HR2018JAMES}. To generalize the relationship between vertical velocity and resolution, let $\alpha$ refer to the ratio of $W_0$, the vertical velocity scale of some reference grid spacing $\Delta x_0$, to $W$, the vertical velocity scale for any $\Delta x$. A power law for $\alpha^{-1}$ in $\Delta x$ is then,
\begin{equation}
\alpha^{-1} = \frac{W}{W_0} = \left( \frac{\Delta x}{\Delta x_0} \right)^n, \label{eq:alpha}
\end{equation}
where $n$ is the power law exponent. 

\cite{RETAL2016CD} derive an estimate $n= h-1$ in equation~\ref{eq:alpha} by combining a scale analysis of the continuity equation with a power law representation $\Delta x^{2h}$ of the second-order structure function of the horizontal wind. Strictly speaking, $\Delta x$ here refers to the distance between two points for which the velocity increment is computed in the structure function, but with this distance set to the model grid-spacing. Regional models analyzed in \cite{RETAL2016CD} provide evidence for $h<1$, implying $n<0$ and consistent with the increase in $W$ with resolution in models.

Observations show that $h=\frac{1}{3}$ for horizontal scales on the order of $100$ km and less \citep[hereafter referred to as the {\em{mesoscale}};][]{L1999JFM,CL2001JGR}. This value for $h$ is also supported by the slope of the kinetic energy spectrum $-\beta$, which can be related to the second-order structure function through the Weiner–Khinchine theorem, $\beta = \left( 2h+1 \right)$, valid in the range $1<\beta<3$ \citep[satisfies stationarity;][]{DETAL1996JAS}. For $h=\frac{1}{3}$, $\beta=\frac{5}{3}$, which is true for the kinetic energy spectrum in observations \citep{NG1985JAS,CETAL1999JGR} and models at the mesoscale \citep[e.g.,][]{THO2006GRL,SPKS2014JAS}. \cite{RETAL2016CD} propose that the overwhelming support for $h=\frac{1}{3}$ at the mesoscale provides an emergent constraint for $n=-\frac{2}{3}$ in equation~\ref{eq:alpha}.

In large-eddy simulations, the sensitivity of vertical velocities to resolution is adequately explained by a scale analysis of the {\color{red}{dry equations of motion}} \citep{WETAL1997MWR,PG2006JAS,JR2016QJRMS}. For hydrostatic scales relevant to AGCMs, a scale analysis of the Poisson equation gives $W \propto D^{-1}$, where $D$ is the horizontal scale of buoyancy perturbations driving vertical motion \citep{HR2018JAMES}. In CAM aqua-planet simulations, the largest source of buoyancy is from stratifrom cloud formation, which are grid-limited with horizontal scales set by the effective resolution of the model \citep[i.e., some multiple of $\Delta x$;][]{S2011LNCSE}, indicating $n=-1$ in equation~\ref{eq:alpha} \citep{HR2018JAMES}. \cite{HR2017JCLIM} has shown that the $n=-1$ scaling does not explain the behavior of CAM4 in a convergence experiment, but follow-up studies \citep{HR2018JAMES,HETAL2019JAMES} indicate that the inadequacy of the $n=-1$ scaling is not definitive, due to large time-truncation errors associated with fixing the physics time-step ($\Delta t_{phys}$) across resolutions in that study.

Another robust response of the CAM/CCM lineage to resolution is an increase in stratiform precipitation rates at the expense of parameterized convective precipitation rates. Stratiform precipitation is sometimes referred to as grid-scale precipitation because unlike parameterized convection, it condenses moisture locally without any transport by unresolved eddies. The resolution dependent partitioning between the two different precipitation routines is shown in Figure~\ref{fig:cam-history}, which is a bar-graph of the climatological, global mean stratiform and convective precipitation rates in prior CAM/CCM convergence studies. The tendency for precipitation rates to shift from the convection scheme to the stratiform scheme with resolution has been documented in other models \citep{PS2002CD,HETAL2006JC,RETAL2016CD,TETAL2018CD}, but none have provided a convincing explanation for this sensitivity. The studies of \cite{KW1991JGR}, \cite{WETAL1995CD} and \cite{W2013QJRMS} indicate that the practice of reducing $\Delta t_{phys}$ with resolution should by itself reduce the convective precipitation rates, however Figure~\ref{fig:cam-history} (top row) indicates that convergence studies with fixed $\Delta t_{phys}$ still show a reduction in convective precipitation rates with resolution.

\begin{figure}[t]
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/cam-history.pdf}\\
\end{center}
\caption{Bar-graph of the convective (solid) and stratiform (white) climatological precipitation rates in prior CAM/CCM convergence studies. Each window contains a single convergence experiment, with identical x-axis; the approximate grid resolution. Colors indicate the model configuration; January ensemble (black), aqua-planet configurations with SST profiles $QOBS$ (blue) and $CNTL$ (red) in \cite{NH2000ASL} and annual means from an AMIP configuration \citep[green;][]{AMIP1999}. Studies included in this figure are \cite{KW1991JGR} (CCM1), \cite{WETAL1995CD} (CCM2), \cite{W2008TELLUS} (CAM3), \cite{RETAL2013JCLIM,ZetAl2014JCb,HR2017JCLIM} (CAM4), \cite{ZetAl2014JCb} (CAM5) and this study (CAM6). {\color{red}{Two additional studies are included that do no not use the NCAR models, \cite{PS2002CD} (HadAM3) and \cite{HETAL2006JC} (ECHAM5) in order to illustrate similarities in resolution sensitivity across different AGCMs.}} The asterisk (*) indicates that parameters are modified with resolution to reduce the dependence of cloud fraction and top-of-atmosphere radiative balance on resolution.}
\label{fig:cam-history}
\end{figure}

In this study, a convergence experiment using CAM, version 6 (CAM6; \url{https://github.com/ESCOMP/CAM}) is carried out and analyzed in detail. It is shown that the resolution sensitivity of vertical velocities are well described with $n=-1$ in equation~\eqref{eq:alpha}, provided $\Delta t_{phys}$ is defined in a way that avoids large truncation errors across resolutions. The increase in stratiform precipitation rates with resolution are attributed to an increase in resolved moisture fluxes at cloud base, primarily due to the increase in upward vertical velocities with resolution. {\color{red}{This increase in ascending motion with resolution drives both greater condensational heating and more frequent and expansive subsidence drying, stabilizing the atmosphere and reducing the activity of the convection scheme and associated precipitation rates with increasing resolution.}} The feedback of the resolved vertical motion on the model physics indicates that the root cause of resolution sensitivity in CAM arises from the sensitivity of resolved dynamical modes to native grid resolution. Section 2 describes CAM6 and the details of the convergence experiment. Section 3 contains a thorough analysis of the CAM6 simulations and Section 4 provides some discussion and conclusions.
 
 \begin{table*}
 \caption{Experimental design and global mean climatologies. $\Delta x$ refers to the average equatorial grid-spacing.}
 \centering
 \scriptsize
 \begin{tabular}{lcccccc}
   \hline
   Variable & $ne20$ & $ne30$ & $ne40$ & $ne60$ & $ne80$ & $ne120$ \\ 
   \hline
   $\Delta x$ (km) & 166.8 & 111.2 & 83.4 & 55.6 & 41.7 & 27.8 \\
   $\nu$ ($m^4/s$) & $1.5 \times 10^{15}$ & $4.0 \times 10^{14}$ & $1.5 \times 10^{14}$ & $4.0 \times 10^{13}$  & $1.5 \times 10^{13}$ & $4.0 \times 10^{12}$\\
    $\Delta t_{phys}$ (s) & 2700 & 1800 & 1350 & 900 & 675 & 450 \\
   Total Cloud Fraction & 0.844 & 0.835 & 0.824 & 0.810 & 0.804 & 0.800 \\ 
   Total Precipitable Water (mm) & 23.31& 23.01 & 22.62 & 22.25 & 21.93 & 21.72 \\
   Convective Precipitation, $P_c$ (mm/day) & 1.91 & 1.83 & 1.68 & 1.47 & 1.29 & 1.08 \\
   Stratiform Precipitation, $P_s$ (mm/day) & 1.26 & 1.42 & 1.60 & 1.85 & 2.05 & 2.22 \\      
 \hline
 \end{tabular}
 \label{tbl:table1}
 \end{table*}
 
\section{Methods}

\subsection{Dynamical Core}

This study uses the spectral-element dynamical core option of Community Atmosphere Model \citep[CAM-SE;][]{DetAl2012IJHPCA}, coupled with a mass conserving, semi-Lagrangian advection method for accelerated multi-tracer transport \citep[CSLAM;][]{LTOUNGK2017MWR}, and dry-mass vertical coordinate with comprehensive treatment of moisture and energy \citep{LetAl2018JAMES}. The dry dynamics are solved using the high-order, momentum, mass and energy conserving spectral element method \citep{TF2010JCP}, with the elements defined by a cubed-sphere grid. The notation for horizontal grid resolution is an `$ne$' followed by the number of elements making up an edge of one cubed-sphere face, e.g., $ne30$. Hyper-viscous $\nabla^{4}$ damping is applied to temperature, dry pressure thickness, rotational and divergent winds \citep{LetAl2018JAMES}. CSLAM tracer transport uses a finite volume grid constructed from the cubed-sphere of elements, and contains the same degrees of freedom as the dry dynamics.

\subsection{Physical Parameterizations}

The physics are evaluated on the finite-volume CSLAM grid, and the tendencies mapped back to the spectral element grid. The coupled system, referred to as CAM-SE-CSLAM, conserves energy, mass and preserves linear correlations between two reactive species to within machine precision \citep{HL2018MWR}. A coarser physics grid, containing $\frac{5}{9}$ fewer degrees of freedom than the dynamical core grid is also available as part of the CAM-SE-CSLAM package \citep{HETAL2019JAMES}. This lower-resolution physics grid is used in this study, but only as a member of a perturbed parameter ensemble and not in the control convergence experiment. The dynamics time-step is subcycled within a longer physics time-step $\Delta t_{phys}$, and the temperature and momentum increments from the physics are divided by the number of subcycles and added to the dynamical core at the beginning of each subcycle. The full moisture increment from the physics is applied only at the start of the first subcycle to conserve tracer mass \citep[$ftype=2$ option in][]{LW2019JAMES}.

The simulations use the CAM6 physics package. The Cloud Layers Unified by Binormals \citep[CLUBB][]{GETAL2002JAS,BOG2013JCLIM} is an assumed filtered density function \citep{G1992JFM} high-order closure model that handles shallow convection, planetary boundary layer mixing and cloud macrophysics. The {\color{red}{macrophysics}} are coupled with a two-moment bulk cloud microphysics scheme with prognostic precipitation \citep{MG2}, and microphysics are coupled with the three mode Modular Aerosol Model \citep{MAM}. The combined macrophysics/microphysics routines generate stratiform clouds and stratiform precipitation. Deep convection is parameterized using a quasi-equilibrium mass flux scheme \citep{ZM1995AO} and an entraining plume calculation \citep[referred to as the dilute convective available potential energy, or {\em{dilute CAPE}} hereafter;][]{RB1992JAS, NRJ2008JC} is used as a convective trigger (convection occurs if dilute CAPE $\geq 70$ J/kg), and for closing the mass fluxes in the cloud ensemble. The deep convection scheme also parameterizes convective momentum transport \citep{RR2008JC}.

\subsection{Experimental Design},\label{sec:ed}

The convergence experiment is performed in an aqua-planet configuration \citep{NH2000ASL,MWO2016JAMES}, an all ocean planet with fixed, zonally symmetric sea surface temperatures modeled after present day Earth \citep[$QOBS$ in][]{NH2000ASL}. The aqua-planets are in a perpetual equinox, and aerosols are largely absent from the simulations. Each simulation is {\color{red}{run}} for one simulated year. Six different horizontal grids are used in this study, which are provided in Table~\ref{tbl:table1}. In addition to the six simulations used in the convergence experiment, an ensemble of 24 simulations containing different model parameters (e.g., using the lower resolution physics grid) and across different resolutions are {\color{red}{run}} for one year in order to increase confidence in assessing resolution sensitivity in this study. All analyses exclude the first month of the simulations, which is sufficient to avoid the spin-up period; the model comes into global energy balance after about ten days (not shown). All calculations are performed on their native grids unless otherwise stated.

The horizontal hyper-viscosity operators $\nu$ vary with resolution after \cite{HETAL2019JAMES}, also provided in Table~\ref{tbl:table1}. The values of $\nu$ are a factor 2.5 greater for divergence damping and are not shown. $\Delta t_{phys}$ is chosen to scale with resolution, in proportion to the grid spacing,
\begin{equation}
\Delta t_{phys} = \Delta t_{phys,0} \times \frac{n_{e,0}}{n_e},\label{eq:dt-scale}
\end{equation}
where $\Delta t_{phys,0}$ is taken to be the standard $1800$ s used in CAM-SE-CSLAM for the standard climate resolution, $n_{e,0} = 30$ (equivalent to an average equatorial grid spacing $\Delta x = 111.2$km). This scaling was chosen to avoid large time-truncation errors in a rising moist bubble test \citep[Appendix A in][]{HETAL2019JAMES}, and it is understood that this choice of $\Delta t_{phys}$ will likely lead to greater resolution sensitivity \citep{W2008TELLUS}. The convective time-scale in the deep convection scheme is fixed at 3600 s in all simulations. {\color{red}{Due to the large sensitivity of AGCM solutions to $\Delta t_{phys}$ \citep{WO2003QJR,W2013QJRMS,HR2018JAMES}, the lack of scientific consensus on the numerical and physical justifications for how $\Delta t_{phys}$ should vary with resolution remains a severe limitation to characterizing solutions across resolutions in AGCMs.}}

 \section{Results}

Table~\ref{tbl:table1} provides some globally averaged, climatological metrics for the CAM6 convergence experiment, commonly published in CAM/CCM convergence studies. Total precipitable water, total cloud fraction and deep convective precipitation rate decreases, while stratiform precipitation increases, monotonically with resolution (also shown in Figure~\ref{fig:cam-history}). Resolution sensitivity in CAM6 is similar to all prior versions of the model. 

\subsection{Vertical Velocities and Resolution}

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_2pdf.pdf}\\
\end{center}
\caption{Probability density distribution of the upward vertical pressure velocities $\omega$ computed everywhere in the model from six-hourly output over the entirety of the year-long simulations. (a) Values on their native grid (solid) and values bilinearly remapped to the $ne20$ grid (dotted), (b) values on their native grid, scaled to the $ne120$ resolution using a power law exponent $n=-1$ in equation~\ref{eq:alpha}.}
\label{fig:2pdf}
\end{figure}

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_diags_2panel.pdf}\\
\end{center}
\caption{Components of the climatological, global mean vertical pressure velocity, (a) $\langle f_{u} \rangle \langle \omega_{u} \rangle$ and (b) $\langle f_{d} \rangle \langle \omega_{d} \rangle$. Grey crosses are for the 24 member perturbed parameter ensemble.}
\label{fig:2panel}
\end{figure}

The probability density function (PDF) of negative, or upward vertical pressure velocities $\omega$ in the aqua-planets is provided in Figure~\ref{fig:2pdf}a. The magnitude of upward $\omega$ increases monotonically with resolution, with positive, or downward $\omega$ behaving similarly (not shown). This monotonic increase in the magnitude of $\omega$ is evident even after remapping all model output to a common grid ($ne20$; dotted curves in Figure~\ref{fig:2pdf}a).

The PDF's may be scaled to the highest-resolution, $ne120$ grid through $ \alpha P (\omega / \alpha)$, where $P(x)$ is the PDF and $\alpha$ the scale factor from equation~\ref{eq:alpha} with $\Delta x_0$ set to the $ne120$ grid-spacing. Figure~\ref{fig:2pdf}b shows the scaled PDF's for a power law exponent $n=-1$ in $\Delta x$. The scaled PDF's all collapse onto the high-resolution reference, indicating that the power-law exponent $n=-1$ explains to first-order the variation in vertical velocity with resolution as shown by the aqua-planet simulations. In contrast, the $n=-\frac{2}{3}$ slope proposed in \cite{RETAL2016CD} greatly underestimates the increasing occurrence of large magnitude vertical velocities with resolution (SFigure 1). {\color{red}{But the failure of any particular value of $n$ to explain vertical velocities in the simulations should be interpreted cautiously. Due to the large sensitivity of vertical velocity to $\Delta t_{phys}$ \citep{HR2018JAMES}, the $n=-1$ exponent may only characterize simulations that use equation~\ref{eq:dt-scale} to define $\Delta t_{phys}$ across resolutions.}}

Changes to the vertical velocity field can be further understood through decomposing the mass weighted vertical mean $ \langle \omega \rangle$ into upward and downward components,
\begin{equation}
\langle \omega \rangle =\langle f_{u} \rangle \, \langle \omega_{u} \rangle + \langle f_{d} \rangle \, \langle \omega_{d} \rangle, \label{eq:omega}
\end{equation}
where $\langle f_x \rangle$ and $\langle \omega_x \rangle$ refers to the vertical mass fraction $ \left( \frac{\int dp_x}{\int dp} \textrm{, with pressure } p \right)$ and the $x$ component of the mass weighted vertical mean of $\omega$ $ \left( \frac{\int \omega_x dp_x}{\int dp_x} \right)$, respectively, subscript $u$ refers to upward motion and $d$, downward motion. {\color{red}{ This breakdown of $\langle \omega \rangle$ into subsiding and ascending components is useful for characterizing the strength of subsidence or ascent in the model, since $\langle \omega \rangle$ includes cancellations in the vertical integral if there is both subsiding and ascending motion in a column.}}

The global mean, climatological components $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ and $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ are provided in Figure~\ref{fig:2panel}a,b for the aqua-planet simulations. The magnitude of both $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ and $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ increase monotonically with resolution, and are equal and opposite, which is a requirement of mass conservation in the model and a convenient check of the calculation. While $\langle f_{d} \rangle$ is about 25\% larger than $\langle f_{u} \rangle$ in all simulations, the vertical mass fractions vary by only few percent with resolution, and so the monotonic behavior of $\langle f_{x} \rangle \, \langle \omega_{x} \rangle$ with resolution is primarily from variations in $ \langle \omega_{x} \rangle$ (not shown).

\subsection{Vertical Velocities and Stratiform Precipitation}

\cite{RETAL2016CD} proposed an approximate scaling for the total precipitation rate in models $P_{tot}$, proportional to the upward moisture flux through cloud base,
\begin{equation}
P_{tot} \approx -\frac{1}{g\rho_{w}} \omega q |_{cb} \label{eq:mflux}
\end{equation}
where $\omega q |_{cb}$ refers to the product of $\omega$ and specific humidity $q$ for ascending motion at cloud base, respectively, with $g=9.80616$ m/s$^2$ the acceleration of gravity and $\rho_w=1000$ kg/m$^3$ the density of rainwater used in CAM. \cite{OETAL2016JAMES} deem this the ``what goes up, must go down" model, highlighting the physically intuitive nature of the scaling, but note the scaling omits other potentially important processes, such as detrainment of cloudy updrafts into the environment or updrafts embedded within efficient circulations, with large gross moist stabilities.

Through approximating the cloud base as the $850$ hPa level, equation~\ref{eq:mflux} was found to provide a good fit to total precipitation rates in a regional model \citep{RETAL2016CD} and in the CAM-SE AGCM with CAM5 physics, across multiple resolutions \citep[recall that CAM-SE is different from CAM-SE-CSLAM, used in this study;][]{OETAL2016JAMES}. As noted by \cite{RETAL2016CD}, using only resolved quantities in equation~\ref{eq:mflux} omits the sub-grid scale contribution to moisture fluxes, and so it is instructive to analyze the relative skill of this approximation for the individual components of $P_{tot}$, being the sum of stratiform $P_s$ and parameterized deep convective $P_c$ precipitation rates. Figure~\ref{fig:mflux} shows the median $P_c$, $P_s$ and $P_{tot}$ conditioned on the $850$ hPa resolved moisture flux scaling using $1$ mm/day bins, in the CAM6 $ne30$ simulation. The figure shows that the moisture flux scaling is a good first-order approximation to the median total precipitation rate, and that the increase in precipitation rates with moisture flux is primarily from the stratiform precipitation scheme. The scaling is just as skillful for different resolutions \citep[not shown;][]{OETAL2016JAMES} and so changes in stratiform precipitation rate with resolution can be understood through quantifying changes in resolved moisture fluxes with resolution.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_mflux.pdf}\\
\end{center}
\caption{Precipitation rates vs. upward moisture flux at the 850 hPa level. Solid lines refer to median stratiform (green), convective (blue) and total (magenta) precipitation rates conditional on bins of the moisture flux, and shaded regions refer to the conditional interquartile ranges.}
\label{fig:mflux}
\end{figure}

 \begin{table}
 \caption{Fractional contribution of latitude bands $\pm 10^{\circ}$ and $\pm 15^{\circ}$ to changes in global mean stratiform and parameterized convective precipitation with resolution. The grid headers refer to differences with respect to the next lowest grid resolution, e.g., $ne30 = ne30-ne20$, $ne40=ne40-ne30$, etc... All differences are computed after conservative remapping to a common $ne20$ grid.}
 \centering
 \scriptsize
 \begin{tabular}{lcccccc}
   \hline
   Variable & $ne30$ & $ne40$ & $ne60$ & $ne80$ & $ne120$ \\ 
   \hline
   $\pm 10^{\circ}$ ($17.6\%$ of global area) \\
   Convective Precipitation, $P_c$ & -0.58 & 0.62 & 0.66 & 0.72 & 0.70 \\
   Stratiform Precipitation, $P_s$ & 0.55 & 0.63 & 0.69 & 0.67 & 0.41 \\ 
   \hline
   $\pm 15^{\circ}$ ($25.8\%$ of global area) \\
   Convective Precipitation, $P_c$ & 0.22 & 0.75 & 0.73 & 0.79 & 0.72 \\
   Stratiform Precipitation, $P_s$ & 0.46 & 0.64 & 0.71 & 0.70 & 0.49 \\      
 \hline
 \end{tabular}
 \label{tbl:table2}
 \end{table}

To unravel the contributions of changes in $\omega$ and $q$ at the 850 hPa level, $\omega_{850}$ and $q_{850}$, to the increase in global mean, climatological stratiform precipitation rates $\overline{P_s}$, $\overline{P_s}$ is decomposed in $\omega_{850}$ and $q_{850}$ space following \cite{TETAL2018CD}. Table~\ref{tbl:table2} shows the fractional contribution of deep tropical belts to the global mean change in stratiform precipitation with resolution. The table indicates that roughly a majority ($45-70 \%$) of the increase in stratiform precipitation with resolution is from changes within the $\pm 15^{\circ}$ latitude band, and so the moisture flux decomposition is restricted to this region to simplify the analysis; $\overline{P_s}$ is redefined as the average over $\pm 15^{\circ}$ latitude. 

$\overline{P_s}$ can be expressed as the double sum of the product of the time mean magnitude $M_s$ and time mean spatial frequency $f_s$, over $\omega_{850}$ and $q_{850}$ space,
\begin{equation}
\overline{P_{s}} = \sum_i \sum_j f_s \left( \omega_i , q_j \right) M_s \left( \omega_i , q_j \right) \label{eq:pdecomp}
\end{equation}
where the subscript $850$ is dropped from $\omega$ and $q$ for brevity. $f_s$ is a measure of the occurrence of a particular combination $\left( \omega_i , q_j \right)$ in a simulation, and $M_s$ the mean stratiform precipitation rate associated with the combination $\left( \omega_i , q_j \right)$. Their product $f_s M_s$ is then the contribution of the combination $\left( \omega_i , q_j \right)$ to $\overline{P_s}$.

Figure~\ref{fig:pdecomp} shows plots of the terms $M_s$ and $f_s M_s$ for all resolutions. The plots are computed using 6-hourly instantaneous output of $\omega_{850}$ and $q_{850}$, with 0.05 Pa/s and 0.4 g/kg bin widths, respectively. Bins in  $\left( \omega_{i} , q_{j} \right)$ space with $f_s <1 \times 10^{-5}$ are masked out, which is a somewhat arbitrary, but reasonable cut-off for a bins' contribution to $\overline{P_s}$ ({\color{red}{all unmasked values of $M_s$ and $f_s$ are plotted in SFigure~2}}). The $M_s$ plots (left column) show that larger magnitude $\omega_{850}$ correspond to larger magnitude stratiform precipitation rates, while the impact of changing $q_{850}$ on $M_s$ is less clear. The changes in $M_s$ with resolution are subtle, while the changes in $f_s$ with resolution are large (not shown). The changes to $f_s$ can be inferred from the larger space of $\left( \omega_i , q_j \right)$ plotted at higher resolutions, indicating that larger magnitude $\omega_{850}$ values are occurring more frequently at higher resolutions, above the cutoff for plotting. The $f_s M_s$ plots (right column) clearly shows that larger magnitude $\omega_{850}$, and therefore larger magnitude stratiform precipitation rates, are contributing to the increase in $\overline{P_s}$ with resolution. In contrast, smaller magnitude $\omega_{850}$ and hence smaller stratiform precipitation rates contribute less and less to $\overline{P_s}$ with increasing resolution.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_pdecomp.pdf}\\
\end{center}
\caption{Decomposition of the climatological stratiform precipitation rates, averaged over the $\pm 15^{\circ}$ latitude band into $\omega_{850}$ and $q_{850}$ environmental conditions. Left column shows the time mean magnitude term $M\left( \omega_i , q_j \right)$ and the right column is the magnitude term multiplied by the space-time frequency term $f\left( \omega_i , q_j \right) M\left( \omega_i , q_j \right)$. Integrals over $f M$ gives the climatological, area averaged stratiform precipitation rate. Panel labels denote the grid resolution of the model run. {\color{red}{See SFigure~2 for all the unmasked values of $M_s$, and the raw $f_s$ field.}}}
\label{fig:pdecomp}
\end{figure}

\subsection{Vertical Velocities and Deep Convective Precipitation}

While the increase in magnitude of vertical velocities with resolution clearly drives an increase in moisture fluxes and stratiform precipitation rates with resolution, it is not clear why there is a reduction in parameterized deep convective precipitation rates with resolution, in all versions of the model (Figure~\ref{fig:cam-history}). Curiously though, there is an excellent negative correlation (Pearson's R-value = 0.99, N = 27) between the global mean, climatological $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ and a measure of the activity of the \cite{ZM1995AO} deep convection scheme (referred to as the {\em{ZM scheme}} hereafter), global mean, climatological $FREQZM$ (Figure~\ref{fig:corr}). At any grid-point and time-step, $FREQZM$ is a binary variable: 1 if the ZM scheme is active, 0 if it is not. Time mean $FREQZM$ is therefore the fraction of the model time that the ZM scheme is triggered, i.e., dilute CAPE exceeds $\geq 70$ J/kg. The regression indicates that model simulations with greater subsidence also have less convective activity.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_diags_corr.pdf}\\
\end{center}
\caption{Scatter plot of global mean, climatological $\langle f_{d} \rangle \langle \omega_{d} \rangle$ and $FREQZM$, and the fitted linear regression which has a Pearson's R-value = 0.99, using all 27 simulations. Grey crosses are for the 24 member perturbed parameter ensemble runs.}
\label{fig:corr}
\end{figure}

Figure~\ref{fig:4zonal}a,b, shows the zonal mean variations in $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ and $FREQZM$ in the convergence experiment. $FREQZM$ is largest in the $\pm 10^{\circ}$ latitude region, within the Inter-Tropical Convergence Zone (ITCZ), and rapidly decreasing polewards into the subtropics. $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ increases away from the ITCZ region and reaches a maximum at the poleward limit of the Hadley Cell. The increase (decrease) in $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ ($FREQZM$) with resolution is to first-order, independent of latitude.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=14pc,angle=0]{figs/temp_4zonal.pdf}\\
\end{center}
\caption{Zonal mean (a) climatological $\langle f_{d} \rangle \langle \omega_{d} \rangle$ and (b) climatological $FREQZM$. Zonal mean (c) R-values and (d) the shape parameter $b_1$ in the logistic regression.}
\label{fig:4zonal}
\end{figure}

To further understand the relationship between subsidence and activity of the ZM scheme, a logistic regression between $\langle f_{d} \rangle \langle \omega_{d} \rangle$ and $FREQZM$ is performed for each grid column within each of the simulations. Logistic regression uses an iterative method to fit a continuous variable predictor, $x$ to a binary predictand $p$ using the exponential \citep{WILKSBOOK},
\begin{equation}
p = \frac{exp{[b_0 + b_1 x]}}{1 + exp{[b_0 + b_1 x]}}, \label{eq:logreg}
\end{equation}
where $b_0$ and $b_1$ are the shape parameters of the exponential. The predictor is the instantaneous $\langle f_{d} \rangle \langle \omega_{d} \rangle$ of a grid column, and the predictand the binary $FREQZM$. The assumption is then that subsidence is the independent variable, which is reasonable considering the environment of subsiding regions is generally more stable than its surroundings, and the ZM scheme is modulated by the dilute CAPE stability calculation. Grid column regressions that are statistically significant at the $95\%$ level using a log-likelihood test \citep{WILKSBOOK} are retained for analysis. Since the aqua-planets have zonally symmetric boundary conditions, there is a zonally varying structure in the goodness of fit (R-value) and shape parameter $b_1$ (Figure~\ref{fig:4zonal}c,d).

The zonal mean R-values indicate the greatest goodness of fit in the $\pm 10^{\circ}$ latitude band, hereafter referred to as the deep tropics. In this region, the shape parameter $b_1$ is large and negative (Figure~\ref{fig:4zonal}d), consistent with the idea that subsiding motion stabilizes the environment and actively depresses dilute CAPE and the activity of the ZM scheme in the simulations. The shape parameter becomes less negative in the deep tropics with resolution, likely due to the greater magnitude $\langle f_{d} \rangle \langle \omega_{d} \rangle$ with resolution, which requires a lower $b_1$ to predict the binary $FREQZM$. The R-values generally decrease with resolution indicating that there is degradation in the relationship with resolution. The logistic regression is less skillful in the subtropics, with skill declining even further in the mid-latitudes irrespective of resolution (Figure~\ref{fig:4zonal}c).

\subsubsection{Deep Tropics}

Table~\ref{tbl:table2} indicates that a majority ($60-70 \%$) of the reduction in convective precipitation with resolution is from changes within the deep tropics (except in going from $ne20$ to $ne30$, where convective precipitation rates increase, in part due to a wide double-ITCZ in the $ne20$ run that spans outside of $\pm 10^{\circ}$ latitude, but which is contained within $\pm 10^{\circ}$ latitude in the $ne30$ run). Expanding the latitude boundaries marginally to $\pm 15^{\circ}$, roughly $75\%$ of the changes in convective precipitation with resolution occurs in this region (again, ignoring $ne30-ne20$; Table~\ref{tbl:table2}). This trend reflects changes in the partitioning of the ITCZ from convective to stratiform precipitation with resolution. 

The large reduction in convective precipitation with resolution in the deep tropics also happens to be the region where the logistic regression indicates that subsiding motion is most skillful at depressing the activity of the convection scheme. But the change in $FREQZM$ in the deep tropics with resolution is not substantially different from any other region (Figure~\ref{fig:4zonal}b), such as the midlatitudes where the logistic regression indicates a poor relationship between subsidence and $FREQZM$ (Figure~\ref{fig:4zonal}c). The deep tropics is then only unique for its substantially larger change in ZM precipitation per change in $FREQZM$ with resolution.

To characterize the changes to dilute CAPE of subsiding regions in the deep tropics with resolution, temperature and moisture profiles are conditionally sampled depending on whether $\langle \omega \rangle$ is positive or negative, indicating predominantly subsiding or ascending grid columns. {\color{red}{In the lagnuage of equation~\ref{eq:omega}, the use of positive $\langle \omega \rangle$ to characterize a grid column as descending is equivalent to saying the magnitude of $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ exceeds the magnitude of $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ in a grid column, and vice versa for grid columns with negative $\langle \omega \rangle$.}} The time mean temperature and moisture profiles of subsiding and ascending regions are then used to compute the dilute CAPE used in the ZM scheme, offline. Figure~\ref{fig:cape}a shows the dilute CAPE values associated with mean conditions for ascending, descending and all grid columns in the deep tropics, with resolution. Ascending regions are associated with larger values of dilute CAPE ($>180$ J/kg) relative to subsiding regions ($<110$ J/kg), and the dilute CAPE in both regimes decreases monotonically with resolution. 

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_cape.pdf}\\
\end{center}
\caption{(a) Dilute CAPE computed from time mean temperature and moisture profiles of ascending (red), subsiding (blue) and all grid columns (black) in the deep tropics ($\pm 10^{\circ}$ latitude), and (b) space-time weights of ascending (red) and descending (blue) grid columns in the deep tropics. (c) Dilute CAPE computed for ascending/descending grid columns, but using the mean temperature profile for the entire deep tropics, and (d) Dilute CAPE for ascending/descending regions but fixing moisture to the $ne20$ profile. Grey crosses in (a) are dilute CAPE derived from the sum of the products of space-time weights with the dilute CAPE values of ascending/descending grid columns.}
\label{fig:cape}
\end{figure}

Figure~\ref{fig:profiles} shows the time mean temperature and specific humidity profiles of subsiding grid cells in the deep tropics, expressed as anomalies from the mean temperature and specific humidity of the entire deep tropics. The mean profiles of subsiding regions have an anomalous warming layer in the $600-800$ hPa layer and an anomalous moisture deficit throughout the entire column. This warming and drying pattern is consistent with the effects of subsidence, whose motion adiabatically warms the environment while simultaneously advecting drier conditions aloft, downward. Both warming and drying the environment oppose the growth of dilute CAPE through reducing parcel buoyancy; warming the environment relative to the temperature of rising air parcels reduces parcel buoyancy \citep{Z2002JGR}, and mixing drier environmental air into rising air parcels reduces the moisture available to warm parcels through latent heating \citep{RB1992JAS}.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_profiles.pdf}\\
\end{center}
\caption{Time mean (a) temperature and (b) specific humidity profiles of subsiding grid cells in the deep tropics ($\pm 10^{\circ}$ latitude) in the convergence experiment, presented as anomalies from the mean temperature and specific humidity of the entire deep tropics in each simulation.}
\label{fig:profiles}
\end{figure}

The large spread in dilute CAPE between ascending/descending regions is crucial for maintaining the relationship shown by the logistic regression, since the much smaller values of dilute CAPE of subsiding grid columns is required to depress dilute CAPE below the threshold for convection. To unravel the contributions of warming and drying shown in Figure~\ref{fig:profiles} to the large spread in dilute CAPE between the two regimes, dilute CAPE is recomputed using the mean specific humidity of ascending/descending regions in the deep tropics, but setting the temperature profile to the mean profile for the entire deep tropics. Figure~\ref{fig:cape}c shows this influence of changing moisture on dilute CAPE, which to first order, explains the large spread in dilute CAPE between the ascending/descending regions in the deep tropics. Temperature differences between ascending/descending regimes has a smaller, second order influence on the spread in dilute CAPE (not shown).

The space-time weights associated with ascending and descending grid columns in the deep tropics vary drastically with resolution (Figure~\ref{fig:cape}b). The subsiding (ascending) space-time weights change from $0.32$ ($0.68$) in the $ne20$ run, monotonically increasing (decreasing) with resolution to $0.51$ ($0.49$) in the $ne120$ run. The increasing occurrence of stable, subsiding grid columns accounts for about half of the changes in dilute CAPE in the deep tropics with resolution, the other half being due to the systematic reduction in dilute CAPE for both ascending and descending regions (Figure~\ref{fig:cape}a). This breakdown is straightforward to compute since the sum of the products of the space-time weights with their associated ascending/descending dilute CAPE values approximate the dilute CAPE computed from the mean temperature/moisture profiles over the entire deep tropics quite well (compare black and grey crosses in Figure~\ref{fig:cape}a). 

To isolate the relative importance of temperature or moisture on the systematic reduction in dilute CAPE with resolution for both ascending/descending regimes, dilute CAPE is recomputed for all resolutions, by first fixing the moisture profiles to the lowest resolution $ne20$ profiles, and then through fixing only temperature to the $ne20$ profiles. Figure~\ref{fig:cape}d shows the influence of changing temperature profile with resolution, with moisture fixed to the $ne20$ profiles, on dilute CAPE of ascending/descending grid columns. It illustrates that the systematic reduction in dilute CAPE with resolution in both regimes is primarily from changes to the temperature field. Moisture changes with resolution do not contribute to the reduction in dilute CAPE with resolution, with the exception that the 20 J/kg reduction in dilute CAPE of ascending regions between the $ne80$ and $ne120$ runs (Figure~\ref{fig:cape}a) is entirely due to changes in moisture (not shown).

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_dhgt_panel.pdf}\\
\end{center}
\caption{{\color{red}{Climatological change in temperature (left column) and the physics temperature tendencies reflecting the sum of parameterized cloud macrophysics, shallow convection and PBL mixing (CLUBB; right column) between low and high resolution simulations. Panels (a-d) show $ne40 - ne20$ differences for (a,b) ascending grid columns and (c,d) subsiding grid columns, whereas panels (e-h) are the $ne80p - ne40$ differences for (e,f) ascending grid columns and (g,h) subsiding grid columns. Thick black contour is the zero isoline in all panels. For reference, the thick magenta lines in the right column denotes the location of the tropopause in the higher resolution simulations using the lapse-rate WMO definition. Only three of the six simulations are used in this calculation because the CLUBB tendency is not available in the other simulations.}}}
\label{fig:dzonal-hgt}
\end{figure}

{\color{red}{This increase in stability with resolution is evident from plots of the zonal mean temperature change with height in going from $ne20$ to $ne40$ ($ne40-ne20$) and from $ne40$ to $ne80$ ($ne80-ne40$; left column of Figure~\ref{fig:dzonal-hgt}). Anomalous warming occurs in $ne40-ne20$ and $ne80-ne40$ throughout most of the deep tropical troposphere, and cannot be attributed to increasing subsiding motion with resolution as it exists in both ascending ($\langle \omega \rangle<0$; Figure~\ref{fig:dzonal-hgt}a,e) and descending grid columns ($\langle \omega \rangle>0$; Figure~\ref{fig:dzonal-hgt}c,g). This trend of tropospheric warming and greater stability with resolution occurs in the mid-latitudes as well, consistent with the reduction in $FREQZM$ in this region (Figure~\ref{fig:4zonal}b).

The right column of Figure~\ref{fig:dzonal-hgt} shows the change in zonal mean CLUBB temperature tendencies versus height for $ne40-ne20$ and $ne80-ne40$. CLUBB parameterizes cloud macrophysics, shallow convection and PBL mixing, and so the sum of these processes are reflected in the tendencies. Ascending grid columns (Figure~\ref{fig:dzonal-hgt}b,f) show a pattern of increasing CLUBB tendencies in the 200-800 hPa layer (\textit{free troposphere}) with resolution that is absent in descending grid columns (Figure~\ref{fig:dzonal-hgt}d,h). This suggests a relative increase in condensational heating from the increase in upward mass fluxes and stratiform cloud formation (Figure~\ref{fig:mflux}). Greater condensational heating is consistent with the anomalous warming that occurs in both ascending and descending regimes, as it would be expected to spread horizontally over the weak temperature gradient regime of the tropics \citep{SNP2001JAS}, and increase the temperature of the mean state more broadly. 

The high negative correlation between $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ and $FREQZM$ in Figure~\ref{fig:corr} not only reflects the reduction in dilute CAPE due to greater subsiding motion, but also greater ascending motion favoring stronger condensational heating in the free troposphere. In the time-mean global integral, any increase in magnitude of $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ is reflected in the magnitude of $\langle f_{d} \rangle \, \langle \omega_{d} \rangle$ through mass conservation (Figure~\ref{fig:2panel}), and so Figure~\ref{fig:corr} incorporates both the impact of greater magnitude descending and ascending motion on $FREQZM$. 

Using instantaneous zonal mean $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ as a predictor of $FREQZM$ in equation~\ref{eq:logreg} is not expected to be very informative due to gravity wave propagation of heating anomalies over the deep tropics, and a positive correlation between resolved mass flux and ZM precipitation rate (Figure~\ref{fig:mflux}). Indeed, doing so provides low R-values and a shape parameter predicting an increase in $FREQZM$ with increasing ascending motion in the deep tropics (SFigure~3). Outside the tropics, where the deep convection scheme is much less active, there is an increase in condensational heating centered about the mid-latitude jet maximum in $ne40-ne20$ and $ne80-ne40$ for ascending grid columns (Figure~\ref{fig:dzonal-hgt}b,f). The logistic regression shape parameter reverses sign poleward of $\pm 40^{\circ}$ latitude (SFigure~3d), indicating that instantaneous $\langle f_{u} \rangle \, \langle \omega_{u} \rangle$ is negatively correlated with $FREQZM$ in these regions. This suggests that the increase in stability and reduction in $FREQZM$ with resolution in mid-latitudes is in part due to the increase in upward vertical velocities and resulting increase in stratiform cloud formation with resolution.}}

\subsubsection{Subtropics}

Figure~\ref{fig:cape-subt}a shows the dilute CAPE values computed from mean temperature and moisture of subsiding and ascending regions in the $\pm \left( 10^{\circ}-30^{\circ} \right) $ latitude bands, hereafter referred to as the subtropics. The spread in dilute CAPE between ascending and descending grid columns is much smaller than for the deep tropics (Figure~\ref{fig:cape}a), and their dilute CAPE  values vary much less with resolution ($\sim 20$ J/kg, compared with $\sim 80$ J/kg across all resolutions in the deep tropics). Through recomputing dilute CAPE and fixing the temperature or moisture profiles to $ne20$ values, the reduction in dilute CAPE with resolution is attributed to both temperature and moisture, but as with the deep tropics, temperature changes have a larger influence (not shown).

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_cape-subtropics.pdf}\\
\end{center}
\caption{(a) Dilute CAPE computed from time mean temperature and moisture profiles of ascending (red), subsiding (blue) and all grid columns (black) in the subtropics ($\pm \left(10^{\circ}-30^{\circ} \right)$ latitude bands), and (b) space-time weights of ascending (red) and descending (blue) grid columns in the subtropics. Grey crosses in (a) are dilute CAPE derived from the sum of the products of space-time weights with the dilute CAPE values of ascending/descending grid columns.}
\label{fig:cape-subt}
\end{figure}

It is the lack of spread in dilute CAPE between ascending/descending regions that explains the declining skill in the logistic regression in the subtropics (Figure~\ref{fig:4zonal}c). Without a strong dependence of dilute CAPE on ascending/descending regimes, subsidence is not a very skillful predictor of depressing dilute CAPE below the threshold for convection. In contrast to the deep tropics, there {\color{red}{are}} also no significant changes to the occurrence of subsiding grid columns with resolution; space-time weights of ascending/descending motion are more-or-less invariant with resolution (Figure~\ref{fig:cape-subt}b).

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_trans.pdf}\\
\end{center}
\caption{(a,b) Two snapshots of $\omega$ for a longitude-pressure transect at $\sim 18^{\circ}$ latitude in the $ne30$ simulation, overlain by the $0.0075$ kg/m$^2$/s contour of the ZM mass flux delineating the region where the ZM scheme is active. {\color{red}{Magenta contour is the $\omega=0$ Pa/s isoline.}}}
\label{fig:transect}
\end{figure}

To understand why dilute CAPE is less sensitive to subsidence in the subtropics, Figure~\ref{fig:transect} shows two snapshots of $\omega$ in the longitude-pressure plane at  $\sim 18^{\circ}$ latitude in the $ne30$ simulations, overlain by an isoline delineating where the ZM mass fluxes are quite active. The ZM mass fluxes typically only extend up to about the 800 hPa level in this region, which often occurs with appreciable subsiding motion aloft. Though properly a deep convection scheme, the ZM scheme is acting as a shallow convection scheme in this region \citep{TETAL2016AGU}. This shallow convection regime tends to produce light rain, or drizzle, which is a common bias in AGCMs \citep{D2006JCLIM}. Figure~\ref{fig:2zonal}a shows the fraction of ZM precipitation $\leq 5$ mm/day in the simulations, which stubbornly persists at $\sim$ 70\% in the $\pm \left(10^{\circ}-20^{\circ} \right)$ latitude bands, irrespective of resolution.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=14pc,angle=0]{figs/temp_2zonal.pdf}\\
\end{center}
\caption{Climatological zonal mean (a) drizzle fraction and (b) surface latent heat fluxes in the convergence experiment. Drizzle fraction is defined as sum  convective precipitation rates $\leq 5$ mm/day divided by the sum of all convective precipitation rates, computed from 6-hourly instantaneous fields over the duration of the simulation.}
\label{fig:2zonal}
\end{figure}

In the $\pm \left(10^{\circ}-20^{\circ} \right)$ latitude bands there are opposing influences on dilute CAPE; a global maximum in surface latent heat fluxes (Figure~\ref{fig:2zonal}b), influencing the thermodynamic state of boundary layer parcels and increasing dilute CAPE from below \citep{Z2002JGR}, and increasing subsidence (Figure~\ref{fig:4zonal}a), which opposes dilute CAPE from above. The shallow convection regime of the ZM scheme is likely a result of these two opposing influences on dilute CAPE, with large latent heat fluxes increasing dilute CAPE above the threshold for convection, but with subsidence restricting dilute CAPE from becoming much larger than this threshold. This interpretation is supported by the logistic regression, which shows a local minimum in goodness-of-fit in the $\pm \left(10^{\circ}-20^{\circ} \right)$ latitude region (Figure~\ref{fig:4zonal}c), indicating that it is particularly difficult for subsiding motion to depress dilute CAPE below the threshold for convection where the surface latent heat fluxes are large.

\section{Discussion and conclusions}

Establishing a complete understanding of resolution sensitivity in atmospheric general circulation models (AGCMs) is crucial, since inevitable responses to native grid resolution cannot be ignored in the pursuit of physical parameterizations compatible with the highly flexible grid structures now being used to maximize efficiency on modern day supercomputers (i.e., scale-aware physics). This study analyzes a convergence experiment in an aqua-planet configuration using the Community Atmosphere Model (CAM), with the spectral-element dynamical core option coupled to the accelerated multi-tracer transport scheme (CAM-SE-CSLAM), and version 6 physics (CAM6), in order to understand longstanding convergence issues that have persisted within the CAM lineage since its inception (see Figure~\ref{fig:cam-history}). As with just about every version of CAM, in CAM6 the atmosphere becomes drier and less cloudy with increasing resolution, and parameterized convective precipitation decreases while stratiform precipitation increases with resolution. Prior studies have established that the drying and reduction in cloudiness with resolution is a result of increases in the magnitude of vertical velocities with resolution, which more efficiently advects dry air aloft, downward \citep{KW1991JGR,WETAL1995CD,HR2017JCLIM}. This study aims to then understand why and how vertical velocities change with resolution, and why convective precipitation decreases at the expense of stratiform precipitation with increasing resolution.

The vertical velocities are found to fit a power law scaling $\Delta x^n$ with $\Delta x$ the grid spacing and $n=-1$, {\color{red}{although the sensitivity of vertical velocities to choice of physics time-step across resolutions \citep{HR2018JAMES} precludes a more definitive statement on the generality of the $n=-1$ exponent in CAM.}} The $n=-1$ exponent is derived from a scale analysis of the Poisson equation, with pressure gradients scaling like the inverse of the horizontal scale of buoyancy perturbations, $D^{-1}$, driving convergence into the air column and bringing about vertical velocities that also scale like $D^{-1}$. It has been previously shown that $D$ is set by stratiform cloud formation, which collapse to the smallest resolvable feature in the model \citep{HR2018JAMES}, the effective resolution \citep{S2011LNCSE}, and so $D \propto \Delta x$ giving $n=-1$. This scaling is consistent with studies describing the resolution sensitivity of large-eddy simulations, which use similar physical arguments, but adapted for non-hydrostatic scales \citep{WETAL1997MWR,PG2006JAS,JR2016QJRMS}.

The $n=-1$ power law scaling is {\color{red}{not necessarily}} in conflict with the scaling proposed by \cite{RETAL2016CD}, $n=h-1$, with $h$ being half the exponent of the second-order structure function $\Delta x^{2h}$ of the horizontal wind. \cite{RETAL2016CD} propose that since $h=\frac{1}{3}$ is supported by observations and models at the mesoscale, $n=-\frac{2}{3}$ is an emergent constraint on models. This is an intriguing proposal and the authors don't seek to dismiss its potential value. But it is suggested that the overwhelming support for the $h=\frac{1}{3}$ exponent in the second-order structure function is only relevant for horizontal scales on the order of $100$ km and less \citep{L1999JFM,CL2001JGR}. This mesoscale regime is only marginally resolved in the highest resolution simulation $\Delta x=27.8$ km ($ne120$), since CAM-SE-CSLAM has an effective resolution in the range of $5-10\Delta x$ \citep{HETAL2019JAMES}. The nonexistence of $h=\frac{1}{3}$ in the $\Delta x=27.8$ km simulation is verified through analyzing the slope of the kinetic energy spectrum $-\beta$, which is in duality with the second-order structure function through the Weiner–Khinchine theorem. For $h=\frac{1}{3}$, $\beta=\frac{5}{3}$, which is shallower than the slope near where the spectrum approaches the effective resolution, for all grid resolutions (Figure~~\ref{fig:2ke}). There is however, a progressive shoaling of the slope $-\beta$ with increasing resolution (Figure~~\ref{fig:2ke}b). This suggests a transition towards the $-\frac{5}{3}$ slope in the simulations, but seeing this process through would require running the model at higher resolutions than considered in this study.

\begin{figure}
\begin{center}
\noindent\includegraphics[width=20pc,angle=0]{figs/temp_kespectra.pdf}\\
\end{center}
\caption{(a) Kinetic energy spectrum and (b) compensated kinetic energy spectrum at the 200 hPa level in the simulations. The diagonal black line in (a) is a reference spectrum $k^{-3}$, with spherical wavenumber $k$. Compensated kinetic energy spectrum refers to the kinetic energy spectrum multiplied by $k^{-\frac{5}{3}}$, so that horizontal lines indicate a spectrum with a $-\frac{5}{3}$ slope. In (b), thick ``caterpillars" are overlain on each spectrum in the $5-10 \Delta x$ range, encompassing the effective resolution of the model.}
\label{fig:2ke}
\end{figure}

An increase in resolved moisture fluxes through cloud base occurs with increasing resolution, and are balanced by an increase in stratiform precipitation rates. Through decomposing stratiform precipitation rates into the components of the moisture flux, it is shown that the increase in stratiform precipitation rates with resolution is primarily due to the increase in resolved upward vertical velocities with resolution. At higher resolutions, larger stratiform precipitation rates increasingly contribute to the increase in climatological stratiform precipitation rate, with smaller magnitude precipitation rates contributing less. The increasing occurrence of larger magnitude vertical velocities and stratiform precipitation rates at higher resolution is at odds with a similar analysis by \cite{TETAL2018CD} using a similar model. The authors {\color{red}{suggest}} that their use of daily averaged fields obscured the covariance between vertical velocity and moisture (i.e., moisture flux), complicating the analysis.

A strong negative correlation is discovered between the climatological, global mean activity of the \cite{ZM1995AO} deep convection scheme (ZM scheme) and global mean subsiding motion in the simulations. Since the ZM scheme is modulated by an entraining plume calculation, referred to as the dilute convective available potential energy (dilute CAPE), the authors sought to understand the influence of subsidence on dilute CAPE. In the deep tropics ($\pm 10^{\circ}$ latitude), subsiding regions have much smaller dilute CAPE values compared to ascending regions due to the influence of the drier subsiding environment on dilute CAPE. With increasing resolution and in the deep tropics, the occurrence of subsiding motion increases dramatically and at the expense of ascending motion, and the activity of the the ZM scheme decreases overall. This is consistent with the $\Delta x^{-1}$ scaling of the vertical velocities, e.g., a doubling of resolution can generate the same upward mass flux in just half the area. The increased efficiency of upward mass fluxes with resolution suggests that the occurrence of ascending motion can decrease with resolution while still maintaining statistical equilibrium with the large-scale forcing.

The subtropics ($\pm (10^{\circ}-30^{\circ})$ latitude) are uniquely less sensitive to subsiding motion. This is likely due to the the large, global maximum in surface latent heat fluxes in the subtropics, which increases dilute CAPE above the threshold for convection, but which is restricted from growing much larger due to substantial subsiding motion aloft. These opposing influences result in small values of dilute CAPE, and the ZM scheme acts more like a shallow convection scheme, producing light precipitation or drizzle, irrespective of resolution. 

In addition to the increasing occurrence of subsidence grid columns with resolution in the deep tropics, temperature and moisture profiles show a decrease in dilute CAPE for both ascending and descending regimes across resolutions. This reduction in background dilute CAPE is not confined to the deep tropics, but is also evident from the dilute CAPE derived from the time mean temperature and moisture profiles everywhere in the model (not shown) and plays a large role in the global mean reduction in activity of the ZM scheme with resolution. A sensitivity analysis of the dilute CAPE calculation indicates that this change in background stability is primarily a result of temperature changes with resolution. 

{\color{red}{Temperatures generally increase with resolution in the free troposphere as a result of increases in condensational heating and stratiform cloud formation arising from larger magnitude ascending motion. The overall increase in atmospheric stability with resolution then occurs through two mechanisms; the direct impact of larger upward vertical velocities on condensational heating, and the indirect impact of more vigorous ascent driving more frequent and expansive compensating subsidence, drying out the atmopshere. Since an increase in global mean ascent is reflected in the global mean subsidence calculation through mass conservation (they must be both equal and opposite), both means of stabilizing the atmosphere contribute to the high correlation between global mean subsidence and activity of the ZM scheme in CAM.}}

{\color{red}{The reduction in convective precipitation rates and increase in stratiform precipitation rates with resolution is not unique to CAM (see HadAM3 and ECHAM5 in Figure~\ref{fig:cam-history}). The HadAM3 model also gives way to an increase in vertical velocities with resolution \citep[Figure~2 in][]{PS2002CD}, and it is reasonable to assume this is a common feature among AGCMs \citep[e.g.,][]{DETALA2016ACP}. The increase in stratiform precipitation rates with resolution in HadAM3 and ECHAM5 are likely driven in a similar way by the increase in resolved moisture flux at cloud base. In as much as the greater condensational heating and subsidence drying feedback onto their stability based deep convection schemes \citep{GR1990MWR,N1994ECMWF}, understanding resolution in CAM may also help explain resolution sensitivity in HadAM3 and ECHAM5.}}

This study argues that the sensitivity of parameterized convection and stratiform precipitation rates to resolution are a response to the increase in magnitude of vertical velocities with resolution. This is essentially the hypothesis put forth by \cite{HR2017JCLIM} to explain resolution sensitivity, that increased vertical velocities with resolution interacts with other model components, steering the model towards a new mean state. Here, those interactions are documented explicitly. Resolution sensitivity therefore arises from the relationship between vertical velocities and resolution, a well-known scale dependency of the equations of motion \citep{O1981JAS}.

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Funding support for this work was in part provided by the U.S. Department of Energy Office of Science (DE-SC0019459) and the National Science Foundation (AGS1648629 and AGS1830729). Computing and data storage resources, including the Cheyenne supercomputer (doi:10.5065/D6RX99HX), were provided by the Computational and Information Systems Laboratory (CISL) at the National Center for Atmospheric Research (NCAR). Herrington would like to thank Dr. Sultan Hameed for his assistance with the statistical analysis used in this study. Model output used in this study is publicly available from the GitHub repository \url{https://github.com/adamrher/2019-cam-resolution/tree/master/data}.

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