## 1. Introduction

*m*

_{Ω}=

*ρa*

^{2}cos

^{2}

*φ*Ω and

*m*

_{r}=

*ρa*cos

*φu,*are derived from the condition of conservation of mass and the equation of zonal momentum (Peixoto and Oort 1992). When using the traditional approximation (Phillips 1966, 1968), whereby assuming a constant earth's radius

*r*=

*a,*these equations read (neglecting furthermore extraterrestrial torques)

**v**has zonal and meridional components

*u*and

*υ,*Ω is the angular speed of the earth's rotation,

*φ*latitude,

*λ*longitude, and

*ρ*the density of the air. The second terms on the left-hand sides of Eqs. (1) and (2) originate from the nonlinear advections and describe the net fluxes of Ω and relative angular momenta across the considered volume. The pressure torque is represented by ∂

*p*/∂

*λ,*

*F*is the zonal component of the divergence of the stress tensor, and

_{λ}*a*cos

*φ*

*ρF*is the friction torque. The terms involving 2Ω sin

_{λ}*φ*can be considered as the torque due to the Coriolis force, 2Ω sin

*φ*

*υρ,*acting with moment arm

*a*cos

*φ.*It converts, through the meridional flow, relative angular momentum to or from Ω angular momentum and is referred to as the Coriolis conversion. Equatorward (poleward) flow increases (decreases) Ω angular momentum and decreases (increases) relative angular momentum. Note that the Coriolis conversion is proportional to but not equal to the meridional mass transport,

*ρυ.*

*M*

_{Ω}= ∫

_{V}

*m*

_{Ω}

*dV*and

*M*

_{r}= ∫

_{V}

*m*

_{r}

*dV*:

*τ*

_{s}, ∫

_{S}

*dS*denotes the integral over the earth's surface; and ∫

_{V}

*dV*the integral over the volume of the atmosphere.

In the past, interest in the axial angular momentum of the atmosphere has stemmed, to a large extent, from studies of the global budget of the earth system with emphasis on the relationship between changes in the axial component of the atmospheric angular momentum and that of the solid earth (for a review see Rosen 1993). Within this context, only the budget of the global absolute angular momentum, *M _{a}* =

*M*

_{Ω}+

*M*is relevant and there is no need to study the separate budgets (3) and (4).

_{r},The situation changes for the most pronounced planetary-scale modes of the atmosphere. Using an integration with the coupled atmosphere–ocean general circulation model, it is found that one of these modes has essentially only relative angular momentum but no Ω angular momentum, while two other modes have much larger Ω than relative angular momentum (von Storch 1999a, 2000). The mode with large relative angular momentum has large zonal wind anomalies in the tropical troposphere and is also described by Kang and Lau (1994) using observational data. The two modes with large Ω angular momentum operate in the mid- and high latitudes. They are the modeled Antarctic and Arctic oscillations, which were identified as the most pronounced extratropical modes by Thompson and Wallace (2000). The observation that the dominant planetary-scale modes have, to the first order, either relative or Ω angular momentum, was the initial motivation for studying the separate budgets Eqs. (3) and (4).

For the atmosphere as a rotating fluid, the situation becomes complicated. As long as the above considered eastward stress is not located at the equator, it accelerates *M*_{r} and, at the same time, induces an equatorward Ekman transport, resulting in a positive *M*_{r} and cancels, to a yet unknown extent, the direct acceleration by the stress. Thus, the friction torque exerted on a rotating fluid cannot freely change the rotation speed as in the case of a rigid body. Moreover, the torque also affects *M*_{Ω}, which is impossible for a rigid body.

This note aims to understand how the friction and pressure torques change the global Ω and relative angular momenta. For this purpose, Eqs. (3) and (4) are reformulated by decomposing the total Coriolis conversion into parts induced by the geostrophic, the Ekman, and the residual ageostrophic meridional velocities, respectively. The conversion induced by the residual ageostrophic meridional velocity is further quantified in section 3.

## 2. An alternative formulation

*υ*in Eq. (7) can be written as

*υ*

*υ*

_{g}

*υ*

_{e}

*υ*

_{a}

*υ*

_{g}is the geostrophic velocity defined by

*φ*∈ (0,

*π*/2) or

*φ*∈ (−

*π*/2, 0), and

*υ*

_{e}is the ageostrophic Ekman velocity defined by

*φ*∈ (0,

*π*/2] or

*φ*∈ [−

*π*/2, 0). In Eq. (12)

*τ*

_{zx}is the vertical eddy flux of zonal momentum near the lower boundary of the atmosphere;

*υ*

_{a}represents the residual ageostrophic flow

*υ*−

*υ*

_{g}−

*υ*

_{e}. The definition of

*υ*

_{g}and

*υ*

_{e}can be found in, for example, Peixoto and Oort (1992).

*υ*

_{g}is defined for any

*φ*with

*φ*∈ (0,

*π*/2) or

*φ*∈ (0, −

*π*/2), the Coriolis conversion induced by geostrophic flow

*υ*

_{g},

_{g}, is given by

*local*expression (11) has singularities at the equator and poles, the

*global*expression (13) is well defined.

If the earth's surface would be flat, the Coriolis conversion induced by *υ*_{g} would be zero, reflecting the known fact that the zonally averaged *υ*_{g} exists only between longitudinal barriers. Thus, as _{g} is different from zero only in the presence of mountains.

*υ*

_{e},

_{e}, is given by

_{e}represents the Coriolis conversion induced by the total Ekman transport.

Equations (13) and (14) are particularly useful for estimating the torques exerted on the ocean. Neglect the residual ageostrophic meridional velocity *υ*_{a}. The pressure torque exerted on the oceanic interior can be obtained by replacing *υ* with *υ*_{g} in Eq. (13) (since *υ*_{e} ∼ 0 in the oceanic interior and *υ*_{a} is neglected) and performing the vertical integral from the sea floor to the bottom of the surface Ekman layer. This approximation was used by Ponte and Rosen (1994). The torque exerted by the wind stress at the sea surface, on the other hand, can be obtained by replacing *υ*_{e} with *υ* in Eq. (14) (when neglecting *υ*_{a} and ∫_{λ} *υ*_{g} *dλ* within the Ekman layer) and performing the vertical integral from the sea surface to the bottom of the Ekman layer. Equations (13) and (14) are less useful for estimating *υ*_{e} and *υ*_{g} with nonzero zonal integral are located in the lower part of the atmosphere so that a separation of the two is difficult.

*M*

_{r}and

*M*

_{Ω}:

The new formulations (17) and (18) do not involve additional approximations, but rely only on Eq. (15), which decomposes the total Coriolis conversion

*υ*

_{a}. Since the total mass is conserved,

*υ*

_{a}must transport mass in the direction opposite to the transport induced by

*υ*

_{a}can be understood as the adjustment of the atmosphere to changes in the moment of inertia induced by mass movements associated with

*τ*

_{xx}and

*τ*

_{yx}), one finds

*m*

_{r}**v**)] can only change the global relative angular momentum by initiating a nonzero ageostrophic conversion. This is because the global integral of momentum transport (as the global integral of mass transport) vanishes, but the global integral of Coriolis conversion, which represents a latitudinally weighted mass transport, is generally not zero. The adjustments to the mass transport associated with

_{a}produce, within the mass conversation constraint, changes in surface torques. In this case, the atmosphere exerts rather than receives a torque.

The fact that the new formulations (17) and (18) contain only Coriolis conversions as forcing terms suggests that the angular momentum of a rotating fluid cannot be changed through a direct acceleration of rotation speed by the torques, as in the case of a rigid body. Instead, it is changed by Coriolis conversion with the aid of mass movements.

## 3. The ageostrophic Coriolis conversion C _{a}

_{a}controls whether variations of relative or those of Ω angular momentum are largest. If |

_{a}| is comparable to |

_{a}∼ −(

*M*

_{r}would be much larger than those of

*M*

_{Ω}. On the other hand, if |

_{a}| is much smaller than |

*M*

_{Ω}would be much larger than those of

*M*

_{r}.

_{a}? For a small volume element, the answer to the question is suggested by Eq. (2), which is the sum of Eq. (19) and the balance

The suggestion that one might neglect the ageostrophic Coriolis conversion in the lower atmosphere is quantified using an integration with the ECHAM1/LSG model (von Storch et al. 1997). In Eqs. (17) and (18) _{a} is replaced with the ageostrophic conversion in the upper atmosphere, ^{u}_{a}*M*_{r}(*M*_{Ω}) and the corresponding forcing −^{u}_{a}^{u}_{a}^{u}_{a}*υ*^{u}, since both the Ekman velocity *υ*^{u}_{e}*υ*^{u}_{g}

The low coherence at high frequencies (Figs. 1 and 2) may be induced by the sampling errors. The problem leads to low coherence at extremely low frequencies, which was discussed in von Storch (1999b). Apart from these problems, which are not the concern of this note, a highly significant 90° phase relation is found at intermediate frequencies for *M*_{Ω} and ^{u}_{a},*M*_{r} and −^{u}_{a}_{a} is essentially controlled by the ageostrophic meridional velocity in the upper atmosphere.

## 4. Concluding remarks

The budgets of global relative and Ω angular momenta, *M*_{Ω} and *M*_{r}, are reformulated. The new formulation emphasizes that the angular momentum of a rotating fluid cannot be changed through a direct acceleration of relative rotation speed by the torques. Instead, it is changed by the Coriolis conversion with the aid of mass transports.

According to the new formulation, the relative strength of _{a} controls the budgets of *M*_{r} and *M*_{Ω}. If the amplitude of _{a} is comparable to that of _{a}. Variations of *M*_{r} would be larger that those of *M*_{Ω}. On the other hand, if _{a} can be neglected relative to _{a} induced by the ageostrophic velocity in the lower part of the atmosphere. The main conversion that determines the time rate of change of *M*_{r} and counteracts the effect of

When applying the result of this note to the angular momentum budgets of the ocean, one should be aware of the fact that the ocean exchanges angular momentum with the atmosphere at the sea surface and with the earth at the bottom and lateral boundaries. If the inputted angular momentum from the atmosphere is immediately transferred to the earth, as suggested by a model integration (Ponte and Rosen 1994), one would have _{a} remains the only forcing in Eq. (17) and cannot be neglected. The variations of *M*_{r} would be essentially generated by _{a}, and those of *M*_{Ω} by −_{a}. Amplitudes of variations of *M*_{r} would be comparable to those of variations of *M*_{Ω}. This issue remains to be further studied using improved ocean models.

## Acknowledgments

I thank Dirk Olbers and Joseph Egger for helpful discussions. Thanks also to Ernst Maier-Reimer, who carefully read the paper.

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As Fig. 1, but for squared coherence and phase spectra between *M*_{r} and its approximated forcing −^{u}_{a}^{u}_{a}*M*_{r} is given in von Storch (1999a)

Citation: Journal of the Atmospheric Sciences 58, 14; 10.1175/1520-0469(2001)058<1995:HDFAPT>2.0.CO;2

As Fig. 1, but for squared coherence and phase spectra between *M*_{r} and its approximated forcing −^{u}_{a}^{u}_{a}*M*_{r} is given in von Storch (1999a)

Citation: Journal of the Atmospheric Sciences 58, 14; 10.1175/1520-0469(2001)058<1995:HDFAPT>2.0.CO;2

As Fig. 1, but for squared coherence and phase spectra between *M*_{r} and its approximated forcing −^{u}_{a}^{u}_{a}*M*_{r} is given in von Storch (1999a)

Citation: Journal of the Atmospheric Sciences 58, 14; 10.1175/1520-0469(2001)058<1995:HDFAPT>2.0.CO;2