## 1. Introduction

The angular momentum of an atmosphere with respect to its rotation axis characterizes its rotary inertia and it is a fundamental physical quantity characterizing the general circulation. In the absence of any surface torque and zonal mechanical forcing, the hydrostatic primitive equations conserve the globally integrated axial angular momentum (AAM) [*Thuburn*, 2008] when assuming a constant pressure upper boundary condition [see, e.g., *Staniforth and Wood*, 2003]. The fluid flow solver (also known as the dynamical core) approximating the solution to the hydrostatic primitive equations should therefore ideally also conserve AAM, however, no dynamical core known to the authors conserves AAM to machine precision. For axisymmetric flows, *Hourdin* [1992] derived a vertical discretization that compensates for the lack of AAM in the horizontal discretization. Hyperviscosity operators can be formulated so that uniform rotation is not affected and thereby the operator is not a source/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 core has been found particularly important for modeling superrotating atmospheres such as the atmospheres of Venus and Titan [e.g., *Lebonnois et al*., 2012, hereafter Leb2012]. If the spurious sources/sinks of AAM in the dynamical core are of similar or larger magnitude than the physical torques, the credibility of the simulation is dubious. Leb2012 found that this was the case in Venus/Titan simulations as well as simplified Earth simulations (see Figure 1) when using NCAR's CAM (Community Atmosphere Model) [*Neale et al*., 2010] adapted for the Venus atmosphere and using the finite-volume dynamical core (referred to as CAM-FV) [*Lin*, 2004]. Similarly, *Lee and Richardson* [2010] found that the simulation of the general circulation of Venus's atmosphere varied significantly between different dynamical cores (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] available in the Geophysical Fluid Dynamics Laboratory (GFDL) Flexible Modeling System (FMS)). In particular, it was noted that the damping operators were very different between the dynamical cores. The superior performing model, in terms of credible atmospheric state, conserved AAM very well [*Lee and Richardson*, 2012].

In Earth's atmosphere, the physical sources/sinks of angular momentum are very large. On the resolved scales (part of the dynamical core), there are large mountain torques due to pressure difference across orography. The mountain torques are predominantly eastward in the tropics and westward in the midlatitudes, and this AAM exchange affects the length of day [see, e.g., *Egger et al*., 2007]. On the unresolved scales, the frictional forces such as boundary layer turbulence and drags from breaking gravity waves alter the AAM budget. Due to these large physical sources and sinks (that are not in a similar balance as for Venus and Titan), the lack of conservation of AAM in the dynamical core (when subtracting the mountain torque) is much less apparent.

It is the purpose of this paper to investigate the globally integrated AAM conservation properties of the spectral-element dynamical core (the CAM-SE dynamical core is the continuous Galerkin spectral finite-element dynamical core in NCAR's High-Order Method Modeling Environment (HOMME) [*Dennis et al*., 2005]; referred to as CAM-SE [*Dennis et al*., 2012]) and to investigate how different numerical operators/options available in CAM-SE affect AAM conservation. The CAM-SE dynamical core can be run at different formal orders of accuracy (by varying the order of the polynomial basis functions) and it accommodates two different treatments of vertical advection that are commonly used: the finite difference treatment of vertical advection that conserves angular momentum and total energy [*Simmons and Burridge*, 1981], which will be referred to as *Eulerian* vertical coordinate (hybrid-sigma), and the floating Lagrangian vertical coordinates for which the vertical advection terms are essentially replaced by periodic vertical remapping of prognostic variables from the floating Lagrangian layers to reference Eulerian (hybrid-sigma) vertical coordinates. This remapping also conserves AAM and optionally total energy [*Lin*, 2004]. The effect on AAM conservation by using these different numerical operators is the main topic of this paper. The AAM analysis is detailed in the sense that not only are the total contributions to AAM from the dynamical core and parameterizations separated but also the breakdown into the relative contributions from diffusion operators and the “inviscid” fluid flow solver. The AAM diagnostics are computed consistently inline in the dynamical core at every dynamics time step and fully consistently with the spectral-element method.

The simulations presented here make use of the idealized Earth configuration called Held-Suarez [*Held and Suarez*, 1994]. In this setup, there is no topography and the parameterization suite is replaced by a relaxation of temperature toward a zonally symmetric state and Rayleigh damping of low-level winds to emulate boundary layer friction [*Held and Suarez*, 1994]. This forcing results in a statistical mean state similar to Earth's atmosphere in terms of producing similar time-averaged zonal jet streams and temperature profiles. The only physical source/sink of AAM in this setup is the Rayleigh damping. The absence of mountain torques and other large subgrid-scale torques makes the Held-Suarez test a good test bed for investigating AAM properties of general circulation models developed for Earth's atmosphere.

The paper is organized as follows. In section 2, the formulas and associated nomenclature for the AAM analysis is introduced. The detailed global AAM analysis is presented in section 3 after a description of the exact CAM-SE dynamical core configuration in terms of polynomial order, viscosity coefficients, time steps, etc. We end the paper with a summary and discussion in section 4.