JAMES, VOL. ???, XXXX, DOI:10.1029/,

Held-Suarez simulations with the Community Atmosphere

Model Spectral Element (CAM-SE) dynamical core: a global

axial angular momentum analysis using Eulerian and floating

Lagrangian vertical coordinates

Peter H. Lauritzen1, Julio T. Bacmeister1, Thomas Dubos2, Sebastien

Lebonnois3,4, Mark A. Taylor5

Abstract. In this paper an analysis of the global AAM conservation properties of NCAR’sCommunity Atmosphere Model Spectral Element (CAM-SE) dynamical core under Held-Suarez forcing is presented. It is shown that the spurious sources/sinks of AAM in CAM-SE are three orders of magnitude smaller than the parameterized (physical) sources/sinks.The effect on AAM conservation by changing various numerical aspects of the dynam-ical core (e.g., different vertical coordinates, reduced formal order of accuracy, increaseddissipation and decreased divergence damping) is investigated. In particular it is notedthat changing from Eulerian (hybrid-sigma) to floating Lagrangian vertical coordinatesdoes not alter the global AAM conservation properties of CAM-SE.

1. Introduction

The angular momentum of an atmosphere with respectto its rotation axis characterizes its rotary inertia and itis a fundamental physical quantity characterizing the gen-eral circulation. In the absence of any surface torque andzonal mechanical forcing, the hydrostatic primitive equa-tions conserve the globally integrated axial angular momen-tum (AAM) [Thuburn, 2008] when assuming a constantpressure upper boundary condition [see, e.g., Staniforth andWood , 2003]. The fluid flow solver (a.k.a. the dynamicalcore) approximating the solution to the hydrostatic primi-tive equations should therefore ideally also conserve AAM,however, no dynamical core known to the authors conservesAAM to machine precision. For axisymmetric flows Hourdin[1992] derived a vertical discretization that compensates forthe lack of AAM in the horizontal discretization. Hyper-viscosity operators can be formulated so that uniform ro-tation is not affected and thereby the operator is not asource/sink for AAM for that part of the flow [see, .e.g,Section 3.3.6 in Neale et al., 2010].

Accurate conservation of AAM in the dynamical corehas been found particularly important for modeling super-rotating atmospheres such as the atmospheres of Venus andTitan [e.g., Lebonnois et al., 2012, hereafter referred to asLeb2012]. If the spurious sources/sinks of AAM in the dy-namical core are of similar or larger magnitude than thephysical torques, the credibility of the simulation is dubi-ous. Leb2012 found that this was the case in Venus/Titan

1Climate and Global Dynamics Division, NCAR EarthSystem Laboratory, National Center for AtmosphericResearch, Boulder, Colorado, USA.

2Ecole Polytechnique, UMR 8539, Laboratoire deMeteorologie Dynamique/IPSL, F-94270, Palaiseau, France

3Sorbonne Universites, UPMC Univ Paris 06, UMR 8539,Laboratoire de Meteorologie Dynamique/IPSL, F-75005,Paris, France

4Centre National de la Recherche Scientifique, UMR 8539,Laboratoire de Meteorologie Dynamique/IPSL, F-75005,Paris, France

5Sandia National Laboratories, Albuquerque, USA.

Copyright 2014 by the American Geophysical Union.

simulations as well as simplified Earth simulations (see Fig-ure 1) when using NCAR’s CAM [Community AtmosphereModel; Neale et al., 2010] adapted for the Venus atmo-sphere and using the finite-volume dynamical core [referredto as CAM-FV Lin, 2004]. Similarly, Lee and Richardson[2010] found that the simulation of the general circulationof Venus’s atmosphere varied significantly between differ-ent dynamical cores1. In particular, it was noted that thedamping operators were very different between the dynami-cal cores. The superior performing model, in terms of cred-ible atmospheric state, conserved AAM very well [Lee andRichardson, 2012].

In Earth’s atmosphere the physical sources/sinks of angu-lar momentum are very large. On the resolved scales (part ofthe dynamical core) there are large mountain torques due topressure difference across orography. The mountain torquesare predominantly eastward in the tropics and westward inthe mid-latitudes, and this AAM exchange affects the lengthof day [see, e.g., Egger et al., 2007a, b]. On the unresolvedscales the frictional forces such as boundary layer turbu-lence and drags from breaking gravity waves alter the AAMbudget. Due to these large physical sources and sinks (thatare not in a similar balance as for Venus and Titan), thelack of conservation of AAM in the dynamical core (whensubtracting the mountain torque) is much less apparent.

It is the purpose of this paper to investigate the glob-ally integrated AAM conservation properties of the spectral-element dynamical core2 [referred to as CAM-SE; Denniset al., 2012] and to investigate how different numerical op-erators/options available in CAM-SE affect AAM conserva-tion. The CAM-SE dynamical core can be run at differentformal orders of accuracy (by varying the order of the poly-nomial basis functions) and it accommodates two differenttreatments of vertical advection that are commonly used:the finite-difference treatment of vertical advection that con-serves angular momentum and total energy [Simmons andBurridge, 1981], which will be referred to as Eulerian verti-cal coordinate (hybrid-sigma), and the floating Lagrangianvertical coordinates for which the vertical advection termsare essentially replaced by periodic vertical remapping ofprognostic variables from the floating Lagrangian layers toreference Eulerian (hybrid-sigma) vertical coordinates. Thisremapping also conserves AAM and optionally total energy[Lin, 2004]. The effect on AAM conservation by using thesedifferent numerical operators is the main topic of this paper.

1

X - 2 LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM

Figure 1. Angular momentum diagnostics for CAM-FV in the Held-Suarez setup [data is from Lebon-nois et al., 2012]. First, second, and third column is total angular momentum (Mr+MΩ), time-tendencies

of AAM due to the dynamical core (( dMdt

)dyn) and physical parameterizations ((dMdt

)phys

), respectively,

as a function of time. Note that the spurious source/sinks of AAM from the dynamical core (secondcolumn) are the same order of magnitude as the physical sources/sinks of AAM (third column).

The AAM analysis is detailed in the sense that not only arethe total contributions to AAM from the dynamical core andparameterizations separated but also the breakdown into therelative contributions from diffusion operators and the ‘in-viscid’ fluid flow solver. The AAM diagnostics are computedconsistently in-line in the dynamical core at every dynamicstime-step and fully consistently with the spectral-elementmethod.

The simulations presented here make use of the idealizedEarth configuration called Held-Suarez [Held and Suarez ,1994]. In this setup there is no topography and the param-eterization suite is replaced by a relaxation of temperaturetowards a zonally symmetric state and Rayleigh damping oflow-level winds to emulate boundary-layer friction [Held andSuarez , 1994]. This forcing results in a statistical mean statesimilar to Earth’s atmosphere in terms of producing similartime-averaged zonal jet streams and temperature profiles.The only physical source/sink of AAM in this setup is theRayleigh damping. The absence of mountain torques andother large sub-grid-scale torques makes the Held-Suareztest a good test bed for investigating AAM properties of gen-eral circulation models developed for Earth’s atmosphere.

The paper is organized as follows. In Section 1 the for-mulas and associated nomenclature for the AAM analysisis introduced. The detailed global AAM analysis is pre-sented in Section 3 after a description of the exact CAM-SEdynamical core configuration in terms of polynomial order,viscosity coefficients, time-steps etc. We end the paper witha summary and discussion in Section 4.

2. Method

2.1. Continuous AAM

When choosing the usual spherical coordinate systemthat rotates with the atmosphere and with coinciding rota-tion axes, the global axial angular momentum (AAM) canbe separated into one part (Mr) associated with the relativemotion of the atmosphere with respect to the planets sur-face (a.k.a. wind AAM) and another part (MΩ) associatedwith the angular velocity Ω (= 2π/d, where d is the lengthof the day) of the planet (a.k.a. mass AAM):

M = Mr +MΩ =

∫Dρ u r cos θ dV +

∫DρΩr2 cos2 θ dV,(1)

where r is the radial distance from the center of the planet, ρthe fluid density, u is the zonal velocity component, θ the lat-

itude, λ longitude, dV = r2 cos θ dλ dθ dr is an infinitesimalspherical volume, and D is the global domain. We make theshallow atmosphere assumption and hydrostatic assumptionso r in (1) is replaced with R (mean radius of the planet) anddr = − 1

ρ gdp (g is the gravitational constant), respectively.

In the absence of any surface torque and zonal mechani-cal forcing, the hydrostatic primitive equations conserve theglobally integrated AAM when assuming a constant pressureupper boundary [see, e.g., Staniforth and Wood , 2003]:

dM

dt= 0. (2)

Typically numerical models are divided into a dynamicalcore (dyn) that, roughly speaking, solves the equations ofmotion on resolved scales and physical parameterizationsthat approximate sub-grid-scale processes (phys). There cantherefore be two sources/sinks of AAM:

dM

dt=(dM

dt

)dyn

+(dM

dt

)phys

. (3)

In Held-Suarez configuration(dMdt

)phys

is from simplified

surface drag that acts on the velocity components only. Con-sequently it does not alter MΩ but only Mr. In the Held-Suarez setup the sources/sinks of AAM in the dynamicalcore are due to numerical errors unless explicit or implicitdiffusion is designed to mimic physical drag. In this studywe assume that the dynamical core approximates the solu-tion to the hydrostatic primitive equations and not any sub-grid-scale processes and it should therefore, according to (2),not be a source/sink of global AAM. The spurious contri-butions to AAM should be much smaller than the physicalsources/sinks of AAM:

0 ∼(dM

dt

)dyn(dM

dt

)phys

. (4)

The change of total AAM due to the dynamical core,(dMdt

)dyn

, is decomposed into two components(dM

dt

)dyn

=(dM

dt

)inviscid

+(dM

dt

)diff

. (5)

The first term on the right-hand side of (5) is the tendency ofAAM due to ‘inviscid dynamics’ or more precisely, dynamics

LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM X - 3

Figure 2. First row depicts total or absolute AAM (M , column 1), the Ω AAM (MΩ, column 2), andrelative AAM (Mr, column 3) as a function of time (day 1000 to 1300) for different polynomial ordersfor CAM-SE based on Lagrangian vertical coordinates. Rows two and three show time-tendencies ofAAM due to the dynamical core (( dM

dt)dyn) and physical parameterizations (

(dMdt

)phys

), respectively,

with the same partitioning as row one. Note that the y-axis unit on row one is 1025kgm2 s−1 whereasthe remaining rows are 1.0 × 1019kgm2 s−2. In the embedded plot (row two, column 1) the unit is1.0× 1017kgm2 s−2.

without any explicit diffusion operators, which is accountedfor in the second term. Explicit diffusion in the CAM-SEmodel is fourth-order hyper-viscosity on all prognostic vari-ables as well as additional damping of divergent modes.

2.2. Discretization of total AAM in CAM-SE

CAM-SE is a highly scalable dynamical core based on aequi-angular cubed-sphere tiling of the sphere with elements.Within each element the variables are represented as polyno-mials on a Lobatto-Legendre quadrature (GLL) grid. Hencegradient, curl and other operations can be computed exactlyand then projected on to the basis. The operators are com-patible (also called mimetic) so key discrete operators anddiscrete integrals satisfy continuum properties [Taylor andFournier , 2010]. In particular, mass is conserved to machine

precision. In the version of CAM-SE based on the Simmons

and Burridge [1981] vertical coordinate/advection, total en-

ergy is conserved to time-truncation errors. For a detailed

description of the CAM-SE dynamical core see Neale et al.

[2010].

In CAM-SE the two components of total AAM are dis-

cretized as

Mr =R3

g

ncol∑i

nlev∑k=1

∆pikuik cos(θi)∆Ai, (6)

X - 4 LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM

and

MΩ =ΩR4

g

ncol∑i

nlev∑k=1

∆pik cos2(θi)∆Ai, (7)

respectively, where ∆pik is the pressure level thickness at theGLL point (cell) with index i (ncol is the number of GLLpoints in each model layer) and vertical index k (nlev = 30is the number of vertical levels), and ∆Ai can be inter-preted as the fictitious spherical area associated with GLLpoint i which in CAM-SE is the product between Gauss-Legendre weights and metric terms3 [Dennis et al., 2012].The terms in the AAM budget are computed ‘in-line’ in thecode at every time-step, i.e.

(dMdt

)dyn

is computed at every

Runga-Kutta stage that makes up the dynamics time-step(∆tdyn = 360s). The AAM tendencies over each Runga-Kutta stage are accumulated over each physics time-step(∆tphys = 1800s). For the Lagrangian vertical coordinateversion of CAM-SE the vertical remapping occurs every halfhour. Hence the vertical coordinate ‘floats’ for five dynamicstime-steps .

3. Results

The horizontal resolution in CAM-SE is specified throughthe number of elements (ne) and number of Gauss-Lobatto-Legendre quadrature (GLL) points along the edge of eachelement (np). With np = 4 the prognostic variables arerepresented with degree 3 (= np − 1) polynomials in eachelement. Here we consider ne30np4, ne45np3 and ne90np2configurations that all have the same number of degrees offreedom (= 6× ne2 × (np− 1)2) but decreasing polynomialorders. Fourth-order viscosity is applied to all prognosticvariables. The hyper-viscosity coefficients for fourth-orderviscosity and additional fourth-order divergence damping(ν = 1.0 × 1015 m4/s and νdiv = 2.5 × 1015 m4/s, respec-tively), and time-steps are held fixed for all ne and np set-tings (time-steps are given in Section 2.2). In the upperthree levels of the model (a.k.a. the model sponge) thereis additional Laplacian damping of the prognostic variableswith coefficients increasing from νtop,2×νtop, to 4×νtop to-wards the model top (νtop = 2.5× 105 m2/s).

The vertical resolution is held fixed (nlev = 30), however,as mentioned in the Introduction we consider two differentvertical coordinates or equivalently two treatments of ver-tical advection terms in the prognostic equations referred

to as ‘Eulerian’ and ‘Lagrangian’. The Eulerian configura-tion is based on finite-differences that conserve energy andangular momentum [Simmons and Burridge, 1981]. Thevertical Lagrangian coordinate configuration follows the Lin[2004] approach where the hybrid-sigma vertical coordinatesurface are allowed to float as material surfaces for severalconsecutive time-steps (1800s in this study) whereafter theprognostic variables are remapped back to the hybrid-sigma(Eulerian) coordinate surfaces.

The default polynomial order used for (climate) simula-tion in CAM-SE is degree three (np = 4) with the abovementioned settings for hyper-viscosity operators and time-steps. The preferred choice of vertical coordinate in climatesimulation is the one based on floating Lagrangian levels.We will refer to this configuration as the default configura-tion.

The Held-Suarez forcing simulations were performed for4000 days although we will only show results for shorter timeperiods after the initial spin-up. After the initial spin-up(approximately 100 days) the global AAM diagnostics arestable; for clarity we only show 300 day periods on the Fig-ures. As mentioned above the AAM diagnostics were com-puted at every internal time-step in-line in the source codeusing native surface integration routines to integrate AAMover each element. Hence AAM diagnostics are completelyconsistent in space and time with the dynamical core numer-ical discretization techniques. Common practice, however,is to compute AAM based on temporally averaged outputof prognostic variables (6-hourly or daily) and subsequentperform mathematical operations to derive AAM.

3.1. Conservation of AAM in default setup

The first row in Figure 2 and 3 show the temporal evo-lution (day 1000 to day 1300) of total AAM, wind AAM(Mr) and mass AAM (MΩ) in columns one, two and three,respectively. The AAM for the default setup is shown withsolid green lines (ne30np4Lagrangian) on Figure 2. Asexpected the total AAM is dominated by the mass AAMwhich is about two orders of magnitude larger than thewind AAM. The total AAM oscillates around approximately1.0395×1028 kgm2/s with deviations of about 1025 kgm2/s.Overall these values are similar to real Earth AAM in termsof relative values and amplitude of the temporal variationof AAM [Egger et al., 2007a].

LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM X - 5

Figure 3. Same as Figure 2 but for CAM-SE based on Eulerian (hybrid-sigma) vertical coordinates.

Row two and three of Figure 2 and 3 show the rel-ative contributions to total change in AAM due tothe dynamical core, ((dM

dt )dyn), and parameterizations,((dM

dt )phys), respectively, again separated into total (col-umn one), wind (column two) and mass (column three)AAM. Note that for clarity rows two and three use thesame y-axis scale (max/min values are ±7×1019). Since((dM

dt )dyn) is small a plot with a 100 times smaller range(±7 × 1017) is embedded in the plot. We note that for

the default configuration the spurious sources/sinks oftotal AAM are about three orders of magnitude smallerthan the parameterized (physical) sources/sinks (equa-tion 4). This result is in sharp contrast to the resultsfor CAM-FV shown on Figure 1 where the spurioussources/sinks from the dynamical core were the sameorder of magnitude as the parameterized sources/sinks.

In examining the breakdown of ((dMdt )dyn) into wind

and mass tendencies for AAM (row two, columns twoand three in Figure 2 and 3, respectively), it is observedthat the amplitudes of these are of order 1019, whereas

X - 6 LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM

Figure 4. Same as row two and three of Figure 2 but for the ‘inviscid’ part of the dynamical core solverand explicit diffusion operators, respectively.

the residual is of order 1016. Hence they balance eachother to the per mil. In other words, ((dMr

dt )dyn) com-pensates very well to fluctuations in mass during advec-tion (dMΩ

dt ). Consequently the fluctuations in M comesentirely from the parameterizations (at least up to thethird digit).

By further separating ((dMdt )dyn) into contributions

from ‘inviscid’ dynamics,(dMdt

)inviscid

, and hyper-

viscosity(dMdt

)diff

, similar balances are observed (Fig-

ure 4 and 5). The breakdown of ‘inviscid’ tendencies oftotal AAM into wind and mass AAM tendencies showsthe same balance as for ((dM

dt )dyn) in terms of magni-tudes (row 2 of Figure 2 and 3 are qualitatively indis-tinguishable from row 1 of Figure 4 and 5; the scales on

the y-axis are identical). The diffusion (hyper-viscosity)contributions to the change in AAM is dominated by ve-locity AAM with

(dMr

dt

)inviscid

approximately two or-

ders of magnitude larger than(dMdt

)diff

; note that the

max/min of the y-scale of the embedded plots on Figure4 and 5 are ±1.3×1017 for column one and two, whereasthe range of column three is [−1.2 × 1015, 0.2 × 1015].

In the remainder of this paper we assess how differentoperator and discretization options (available in CAM-SE) affect the conservation of AAM. The impact onconservation of AAM by changing polynomial degree,change vertical coordinate and make other more sub-tle changes in the dynamical core, is discussed in threeseparate sections below.

LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM X - 7

Figure 5. Same as row two and three of Figure 3 but for the ‘inviscid’ part of the dynamical core solverand explicit diffusion operators, respectively.

3.2. Polynomial order and AAM conservation

One may readily pose the question if the AAM con-servation properties are a function of the formal orderof accuracy of the operators employed in the dynamicalcore. Since CAM-SE is based on a Galerkin method, itis rather straight forward to alter the formal accuracyof the horizontal operators by reducing the polynomialorder from four to three (ne45np3) and two (ne90np2).The total number of degrees of freedom held constant(number of elements is increased accordingly), i.e. theresolution for all configurations is approximately 1. Allother parameters are held fixed.

Without loss of generality in this discussion we focuson Figure 2 and 4 (floating Lagrangian vertical coordi-nate solutions). In terms of the time-evolution of totalAAM, the amplitudes of the fluctuations are very simi-lar for the fourth-order (np = 4; green line), third-order(np = 3; red line), and second-order (np = 2; blue line)solutions. There is an approximately 5% relative differ-

ence between the time-mean values of Mr between themodel configurations (Fig. 4; top-middle).

The spurious contributions to AAM from the dynam-ical core is only weakly dependent on polynomial order.The second-order solution (np = 2) has about an or-der of magnitude larger contributions to AAM than thehigher-order solutions. However, the spurious contribu-tions are still two orders of magnitude smaller than thephysical sources/sinks. I.e. the balance between windand mass AAM is still very well maintained despite theformal low order of accuracy. As for the np = 4 results,the change in AAM is mainly due to ‘inviscid’ dynamics(Figure 4).

3.3. Eulerian and Lagrangian vertical advection

Since both the Simmons and Burridge [1981] verticaldiscretization and the floating Lagrangian coordinates[Starr , 1945; Lin, 2004] are available in CAM-SE, itprovides a unique opportunity to compare these verti-cal advection approaches. The simulations with differ-ent order of polynomials were repeated with the floating

X - 8 LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM

Eulerian (hybrid-sigma) vertical coordinate. The AAMdiagnostics are plotted on Figure 3 and 5.

Both vertical advection operators conserve AAM,however, the frequency of the vertical advection step isdifferent between the two. In the Eulerian vertical coor-dinate version the vertical advection is performed at ev-ery dynamics time-step. With the floating Lagrangianvertical coordinate the vertical advection (remapping)is only performed every fifth dynamics time-step.

Perhaps surprisingly we see very little dependency onvertical coordinate and AAM conservation (Figure 2, 3,4, and 5). The main difference is on the embedded plotin Figure 4 and 5 (column three, row two). Since theEulerian vertical coordinate version does not have anyexplicit diffusion on pressure, the mass AAM tendency(dMΩ

dt

)diff

is zero. The floating Lagrangian vertical co-ordinate version of CAM-SE applies explicit diffusion topressure-level thickness and hence the mass tendenciesof AAM associated with diffusion is non-zero.

3.4. Other numerical configuration changes andAAM

A number of experiment were performed altering dif-ferent aspects of the CAM-SE dynamical core, however,none of them changed the total AAM budget signifi-cantly (Figure 6 and 7). The experiment configurationswere as follows:

• five-fold increase in hyper-viscosity coefficient (ν =5.0 × 1015),

• turn off increased divergence damping (νdiv = ν =1.0 × 1015),

• remove uniform rotation correction in the hyper-viscosity computation,

• change from piecewise parabolic method [PPM;Colella and Woodward , 1984] vertical remapping algo-rithm to the piecewise spline method [PSM; Zerroukatet al., 2006],

• turn global total energy-fixer off,

• turn off sponge layer diffusion, i.e. no increasediffusion near model top (namelist variable nu_top

changed from 2.5 × 105 to zero),

• in vertical remapping algorithm (diagnostics notshown on Figure 6 and 7): instead of remapping tem-perature remap total energy and diagnose temperature[Lin, 2004].

4. Summary and discussion

It has been shown using the Held-Suarez idealizedEarth setup that NCAR’s Community AtmosphereModel Spectral Element (CAM-SE) dynamical coreconserves axial angular momentum very well. Thatsaid, even if good global conservation of AAM is demon-strated, it cannot be ruled out that local conservationerrors somehow cancel out. Nevertheless local conser-vation implies global conservation, so any defects in the

global conservation point to deficiencies in local conser-vation.

In CAM-SE the spurious sources and sinks ofAAM from the dynamical core are three orders ofmagnitude smaller than the physical (parameterized)sources/sinks. Changing the vertical coordinate fromthe popular Simmons and Burridge [1981] method tofloating Lagrangian vertical coordinates [Starr , 1945;Lin, 2004] did not affect the global AAM budget. Aslight degradation on the conservation of total AAMwas observed when changing from formally fourth-orderspatial discretization to second-order, however, the spu-rious dynamical core sources/sinks were still two or-ders of magnitude smaller than the change in angularmomentum due to physical forcing. Changing verti-cal remapping algorithm, increasing hyper-viscosity, de-creasing divergence damping, remapping total energyinstead of temperature in the floating Lagrangian ver-tical coordinate version, turning global energy-fixer off,remove uniform rotation correction in diffusion operator(so that uniform rotation is damped), and turning offincreased sponge layer diffusion, did not significantlyaffect the magnitude of the spurious sources/sinks ofglobal AAM.

In the literature excellent AAM conservation proper-ties have been reported in a model based on the spec-tral transform method [Lee and Richardson, 2012]. TheCAM-SE and global spectral transform dynamical coresshare some aspects such as co-location of prognosticvariables (Arakawa A-grid staggering), make use of ba-sis functions so derivative and integrals can be exactlycomputed (except for non-linear terms), and the dif-fusion operators are higher-order (typically forth-orderor higher). Models that have been shown4 (in Venussetup) to have a larger sensitivity to ‘details’ in the dy-namical core are based finite-volume or finite-differencemethods [Lee and Richardson, 2010]. These dynamicalcores are based on staggered grids and have implicit dif-fusion. Note that this does not necessarily imply thatfinite-volume and finite-difference models are not suit-able for simulating super-rotation. For example, theLMD GCM [Hourdin et al., 2006] is capable of simulat-ing super-rotation [e.g. Lebonnois et al., 2010] despitebeing a finite-difference model on a staggered grid.

That AAM conservation is, at least for CAM-SE,not primarily controlled by the vertical discretizationsuggests to look for the cause for good or bad con-servation in the horizontal discretization. Thereforeit would be useful to analyze AAM conservation al-ready at the level of the shallow-water equations. Thesimplicity of the equations may allow a more in-depthunderstanding of the spurious sources of AAM asso-ciated to the various discretization strategies. A firststep towards such an analysis would be the specifica-tion of a shallow-water numerical experiment exacerbat-ing defects in AAM conservation; an aspect currentlyoverlooked by the standard test cases that numerical

LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM X - 9

Figure 6. Same as Figure 4 but for ne30np4 based on Lagrangian vertical coordinates where different as-pects of dynamical core were altered: five-fold increase in hyper-viscosity coefficient (ν = 5.0×1015m4/s;blue line), turn off increased divergence damping (νdiv = ν = 1.0 × 1015m4/2; red line), and removeuniform rotation correction in the hyper-viscosity computation (green line).

Figure 7. Same as Figure 4 but for ne30np4 based on Lagrangian vertical coordinates where differentaspects of dynamical core were altered: change from piecewise parabolic method (PPM) vertical remap-ping algorithm to the piecewise spline method (PSM; dark grey line), turn global total energy-fixer off(orange line), and turn off additional sponge layer diffusion (turquoise line).

schemes are expected to pass [e.g. Williamson et al.,1992].

The authors believe that the global AAM conserva-tion properties of CAM-SE demonstrated in this pa-

per, is an important first step in qualifying CAM-SE

X - 10 LAURITZEN ET AL.: CAM-SE ANGULAR MOMENTUM

for the successful simulation of super-rotating atmo-spheres [Parish et al., 2012]. Contrary to the globalspectral transform models widely used for Venus andTitan modeling, CAM-SE has been demonstrated to behighly scalable on massively parallel compute architec-tures [Taylor et al., 2008]. CAM-SE has also success-fully been used to simulate the Quasi-Biennial Oscilla-tion [QBO Richter et al., 2013].

Acknowledgments. The authors thank two anonymous re-viewers for their reviews. NCAR is sponsored by the NationalScience Foundation (NSF). Partial support for this work was pro-vided through the Scientific Discovery through Advanced Com-puting (SciDAC) program (DE-SC0006745) funded by U.S. De-partment of Energy, Office of Science, Advanced Scientific Com-puting Research. The discussions with J.-F. Lamarque (NCAR)and the software engineering support from B. Eaton (NCAR), S.Santos (NCAR) and S. Goldhaber (NCAR) is gratefully acknowl-edged.

Notes

1. the ‘B-core’ (http://math.nyu.edu/˜gerber/pages/climod/bgrid.pdf),a spectral transform dynamical core [Held and Suarez , 1994],and a finite-volume (FV) dynamical core [Lin, 2004] availablein the Geophysical Fluid Dynamics Laboratory (GFDL) Flex-ible Modeling System (FMS)

2. the CAM-SE dynamical core is the continuous Galerkin spec-tral finite element dynamical core in NCAR’s High-OrderMethod Modeling Environment [HOMME; Dennis et al.,2005]

3. the weights and metric terms are so that∑ncol

i∆Ai equals

the area of the unit sphere (= 4π)4. Lee and Richardson [2010] analyzed simplified Venus simu-

lations with the B-core [Arakawa and Lamb, 1981], a spec-tral transform dynamical core [Held and Suarez , 1994], anda finite-volume (FV) dynamical core [Lin, 2004] available inthe Geophysical Fluid Dynamics Laboratory (GFDL) FlexibleModeling System (FMS)

References

Arakawa, A., and V. R. Lamb (1981), A potential enstrophyand energy conserving scheme for the shallow water equations,Mon. Wea. Rev., 109, 18–36.

Colella, P., and P. R. Woodward (1984), The piecewise parabolicmethod (PPM) for gas-dynamical simulations, J. Comput.Phys., 54, 174–201.

Dennis, A. Fournier, W. F. Spotz, A. St-Cyr, M. A. Taylor, S. J.Thomas, and H. Tufo (2005), High-resolution mesh conver-gence properties and parallel efficiency of a spectral elementatmospheric dynamical core, Int. J. High Perform. Comput.Appl., 19 (3), 225–235, doi:10.1177/1094342005056108.

Dennis, J. M., J. Edwards, K. J. Evans, O. Guba, P. H. Lauritzen,A. A. Mirin, A. St-Cyr, M. A. Taylor, and P. H. Worley (2012),CAM-SE: A scalable spectral element dynamical core for theCommunity Atmosphere Model, Int. J. High. Perform. C.,26 (1), 74–89, doi:10.1177/1094342011428142.

Egger, J., K. Weickmann, and K.-P. Hoinka (2007a), Angularmomentum in the global atmospheric circulation, Reviews ofGeophysics, 45 (4), n/a–n/a, doi:10.1029/2006RG000213.

Egger, J., K. Weickmann, and K.-P. Hoinka (2007b), Angularmomentum in the global atmospheric circulation, Reviews ofGeophysics, 45 (4), n/a–n/a, doi:10.1029/2006RG000213.

Held, I. M., and M. J. Suarez (1994), A proposal for the in-tercomparison of the dynamical cores of atmospheric generalcirculation models, Bull. Amer. Meteor. Soc., 75, 1825–1830.

Hourdin, F. (1992), Conservation du moment cinetiquedans le modele de circulation generale du LMD, Tech.Rep. 175, Laboratoire de Meteorologie Dynamique (LMD),

Paris, France, http://lmdz.lmd.jussieu.fr/developpeurs/notes-techniques/ressources/conserv.pdf.

Hourdin, F., I. Musat, S. Bony, P. Braconnot, F. Codron, J.-L.Dufresne, L. Fairhead, M.-A. Filiberti, P. Friedlingstein, J.-Y.Grandpeix, G. Krinner, P. LeVan, Z.-X. Li, and F. Lott (2006),The LMDZ4 general circulation model: climate performanceand sensitivity to parametrized physics with emphasis on trop-ical convection, cd, 27 (7-8), 787–813, doi:10.1007/s00382-006-0158-0.

Lebonnois, S., F. Hourdin, V. Eymet, A. Crespin, R. Fournier,and F. Forget (2010), Superrotation of venus’ atmosphere anal-ysed with a full general circulation model, J. Geophys. Res.,115 (E06006), doi:10.1029/2009JE003458.

Lebonnois, S., C. Covey, A. Grossman, H. Parish, G. Schubert,R. Walterscheid, P. H. Lauritzen, and C. Jablonowski (2012),Angular momentum budget in general circulation models of su-perrotating atmospheres: A critical diagnostic, J. Geo. Res.:Planets, 117 (E12), n/a–n/a, doi:10.1029/2012JE004223.

Lee, C., and M. Richardson (2012), Angular momen-tum conservation in a simplified venus general cir-culation model, Icarus, 221 (2), 1173 – 1176, doi:http://dx.doi.org/10.1016/j.icarus.2012.10.007.

Lee, C., and M. I. Richardson (2010), A general circula-tion model ensemble study of the atmospheric circulationof Venus, J. Geo. Res.: Planets, 115 (E4), n/a–n/a, doi:10.1029/2009JE003490.

Lin, S.-J. (2004), A ’vertically Lagrangian’ finite-volume dynam-ical core for global models, Mon. Wea. Rev., 132, 2293–2307.

Neale, R. B., C.-C. Chen, A. Gettelman, P. H. Lauritzen, S. Park,D. L. Williamson, A. J. Conley, R. Garcia, D. Kinnison, J.-F.Lamarque, D. Marsh, M. Mills, A. K. Smith, S. Tilmes, F. Vitt,P. Cameron-Smith, W. D. Collins, M. J. Iacono, R. C. Easter,S. J. Ghan, X. Liu, P. J. Rasch, and M. A. Taylor (2010), De-scription of the NCAR Community Atmosphere Model (CAM5.0), NCAR Technical Note, National Center of AtmosphericResearch.

Parish, H., S. Lebonnois, G. Schubert, C. Covey, R. Walterscheid,and A. Grossman (2012), Importance of the angular momen-tum budget in Venus atmosphere general circulation models,Comp. Climatol. Terr. Planets, pp. 1–2.

Richter, J. H., A. Solomon, and J. T. Bacmeister (2013), On thesimulation of the Quasi-Biennial Oscillation in the CommunityAtmosphere Model, Version 5, J. Geophys. Res., submitted.

Simmons, A. J., and D. M. Burridge (1981), An energyand angular-momentum conserving vertical finite-differencescheme and hybrid vertical coordinates, Mon. Wea. Rev.,109 (4), 758–766.

Staniforth, A., and N. Wood (2003), The deep-atmosphere eu-ler equations in a generalized vertical coordinate, Mon. Wea.Rev., 131, 1931–1938, doi:10.1175//2564.1.

Starr, V. P. (1945), A quasi-Lagrangian system of hydrodynami-cal equations., J. Atmos. Sci., 2, 227–237.

Taylor, M., J. Edwards, and A. St-Cyr (2008), Petascale atmo-spheric models for the community climate system model: newdevelopments and evaluation of scalable dynamical cores, J.Phys.: Conf. Ser., 125, doi:10.1088/1742-6596/125/1/012023.

Taylor, M. A., and A. Fournier (2010), A compatibleand conservative spectral element method on unstructuredgrids, J. Comput. Phys., 229 (17), 5879 – 5895, doi:10.1016/j.jcp.2010.04.008.

Thuburn, J. (2008), Some conservation issues for the dynamicalcores of NWP and climate models, J. Comput. Phys., 227,3715 – 3730.

Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N.Swarztrauber (1992), A standard test set for numerical approx-imations to the shallow water equations in spherical geometry,J. Comput. Phys., 102, 211–224.

Zerroukat, M., N. Wood, and A. Staniforth (2006), The parabolicspline method (PSM) for conservative transport problems, Int.J. Numer. Meth. Fluids, 51, 1297–1318.

Corresponding author: P. H. Lauritzen, Atmospheric Mod-eling and Predictability Section, Climate and Global DynamicsDivision, National Center for Atmospheric Research, 1850 TableMesa Drive, Boulder, CO 80305, USA. (pel@ucar.edu)