## 1. Introduction

[2] Numerical simulation of the superrotating atmospheres of Venus and Titan has long been a challenge. Superrotation is present when the total atmospheric angular momentum (AAM) referred to the rotation axis of the planet is larger than the angular momentum the atmosphere would have if it was at rest (no wind relative to the surface). It is directly related to the zonal wind field *u*, the component of the wind that is tangent to latitude circles. Many General Circulation Models (GCMs) of Venus and Titan have reached superrotation, starting from rest, but few have obtained satisfactory agreement with the strong superrotation observed in their atmospheres.

[3] The accuracy of AAM conservation in Earth GCMs has occasionally been studied, e.g. by *Boer* [1990] for the Canadian Climate Center model and by *Lejenäs et al.* [1997] for the US Community Climate Model (CCM). Both studies find that friction and mountain torques account for the simulated AAM variations with very good precision. The CCM is an ancestor of the Community Atmosphere Model (CAM) used for some of the work discussed in this paper. For strongly superrotating atmospheres, several studies indicate the importance of a GCM's angular momentum conservation for producing satisfactory simulations. In the first work to successfully simulate the superrotation in Titan's atmosphere [*Hourdin et al.*, 1995], the authors acknowledge the importance of angular momentum conservation in the GCM and indicate that in their study, seasonal as well as short-term fluctuations of AAM are very close to that computed from the friction torque. Experiments by*Del Genio and Zhou* [1996]using the GISS GCM adapted for rotation rates of Titan and Venus have also indicated the need for accurate angular momentum conservation. Using single or double precision in their computations made a significant difference in the modeled circulations, especially in the case of a Venus-like rotation rate. More recently,*Newman et al.* [2011] have shown that the horizontal dissipation parameterization in the TitanWRF GCM has a strong influence on its ability to spin up Titan's atmospheric circulation.

[4] In recent Venus GCMs, different conclusions were drawn concerning the role of topography, first introduced in *Herrnstein and Dowling* [2007]. A comparative study of Venus GCMs in simplified configurations has been carried out by a working group of the International Space Science Institute (ISSI) in Bern, Switzerland [*Lebonnois et al.*, 2012b].

[5] A precursor comparative study was done by *Lee and Richardson* [2010]. These studies illustrate the wide disparity in results obtained with similar Venus GCMs forced with the same physical parameterizations. All the above work shows how sensitive the circulations in the atmospheres of Venus and Titan are to the numerics in dynamical cores of GCMs.

[6] In this paper, we address the issue of angular momentum conservation in GCMs we have used to simulate the atmospheres of Venus, Earth and Titan. The total AAM of an atmosphere *M* is given by

where Ω is the rotation rate of the planet, *a* is its radius, *g* is its surface gravity, *θ* is latitude, *p*_{s} is surface pressure, *u* is the zonal wind, ∫_{V} *dm* is the integral of mass over the volume of the atmosphere, and ∫_{S} *dS* is the integral over the surface of the planet. The first term *M*_{o} is due to the solid body rotation of the planet (*o* stands for “omega”) and the second term *M*_{r} is due to the movement of the atmosphere relative to the surface (*r* stands for “relative”). Hydrostatic equilibrium is used here, as well as the approximations that are present in the GCMs equations (shallow and spherical atmospheres). The model top is taken at *p* = 0 Pa. *M*_{o} and *M*_{r} are computed in the GCM simulations with discretized sums over the grid cells, with Δ*S* = *a*^{2} cos *θ*Δ*λ*Δ*θ* (where *λ* is the longitude) and Δ*m* = |*Δp*|/*g* × Δ*S*.

[7] In any GCM, the way the AAM evolves during a simulation can be divided into several components. The temporal evolution of the AAM is related to variations in the distribution of the mass at the surface (*dp*_{s}/*dt*, yielding *dM*_{o}/*dt*) or to variations of the zonal wind speed (*du*/*dt*, yielding *dM*_{r}/*dt*). Changes of *M* arise from exchanges of momentum at the surface and at the model's upper boundary, and to conservation errors within the GCM:

where *F* is the total angular momentum tendency from the boundary layer scheme (friction near the surface), *T* is the rate of exchange of AAM with the surface due to surface pressure variations, related to topographical features (mountain torque), *S* is the total angular momentum tendency due to upper boundary conditions (e.g., a sponge layer), *D* is a residual torque due to conservation errors in the parameterization of horizontal dissipation, and *ϵ* is a residual numerical rate of angular momentum variation due to other conservation errors. In the usual GCM configuration, the physical parameterizations are separated from the dynamical core. The *S* and *D* terms are part of the dynamical core though *S* may be in the physical package in some GCMs.

[8] To get an estimate of *ϵ* in a GCM simulation, the different variables in equation (2) need to be computed. Some contributions to the AAM are easy to isolate. In the GCMs used for this study, the only contribution to zonal wind variations computed in the physical package arises in the planetary boundary layer scheme (friction near the surface) . From this tendency, *F* can be computed from

The total contribution from the dynamical core (dycore) to variations in the zonal wind can be obtained at runtime by computing the change in zonal wind between the end of one call to the physical package and the start of the next call. This contribution includes all variations of *u* occurring in the dynamics. From this tendency, the associated torque *Dy* can be computed as

The evolution of the relative atmospheric angular momentum *M*_{r} can then be written as

The mountain torque *T* is included in *Dy* but can also be evaluated separately. Any resolved wave forced by the orography induces surface pressure variations that will then produce a drag through the mountain torque. The mountain torque results from pressure forces from the surface on the atmosphere:

where is the unit vector normal to the surface, and *ds* is the actual surface element. In the spherical referential ( _{r}, _{λ}, _{θ}), *ds* coordinates are *dS* × , where *z*_{s} is the topographic height and *dS* = *a*^{2} cos *θdλdθ* is the horizontal surface element. Therefore, given the approximation = *a* , we have

and projected on the rotation axis ( ):

Subgrid scale gravity waves may also produce an additional drag, but it needs to be parameterized in the physical package to be present and taken into account in the GCMs. An additional term is then added to *F*. This is not the case in our simulations.

[9] Using equation (5) in equation (2) and then solving for *S* + *D* + *ϵ*, the unphysical contribution to the simulated AAM can be estimated from

where the three terms on the right side are evaluated from equations (4), (6) and the time derivative of the *M*_{o} part of equation (1). The integrals are evaluated by summing over the GCM's grid cells. When the tendencies and can be obtained from the dycore during a GCM simulation run, *S* and *D* can be computed individually. However, in some cases it is not possible to easily extract these tendencies due to the structure of the dycore.

[10] In this work, the sources and sinks of AAM variations, including unphysical contributions, are estimated for two different GCMs: the LMD GCM, in its Venus [*Lebonnois et al.*, 2010] and Titan [*Lebonnois et al.*, 2012a] configurations, and the Community Atmosphere Model, in its Venus [*Parish et al.*, 2011] and simplified Earth [*Held and Suarez*, 1994] configurations. These models and the simulations done for this study are described in Section 2. In Section 3, contributions to the AAM variations are illustrated, and consequences of unphysical sources and sinks for the circulation are discussed, especially with regard to the superrotation question.