# Rotational and Divergent Geopotential Components

Kevin E. Trenberth National Center for Atmospheric Research, Boulder, Colorado

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Shyh-Chin Chen Scripps Institution of Oceanography, La Jolla, California

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## Abstract

It has been traditional in meteorology to divide the velocity field up into rotational and divergent components, but not the geopotential field. Yet any balance condition, such as the geostrophic relation or linear balance equation, is a diagnostic relation which states that not only can the balanced velocity field be computed from the geopotential field, but also that the geopotential can be derived from the velocity field. If the latter approach is adopted, then the difference between the observed and computed geopotential is the quantity we refer to as the divergent, or in some cases, ageostrophic, geopotential. In fact, for any balance set of equations, it is essential to partition the geopotential in such a way in order to derive an equivalent set of momentum equations.

The momentum equations equivalent to the linear balance set of equations are given. The linear balance equation is integrated to give a diagnostic relation between the rotational wind components and the gradient of the rotational geopotential plus an extra term which involves the gradient of the planetary vorticity advection potential (PVAP).

The partitioning of the velocity field into rotational and divergent parts v = vr + vd does not depend on any other field. Given vr and the associated streamfunction, we have computed Φ, from the linear balance equation. The difference Φd = Φ − Φr is of order Rossby number times Φ, has global scale dominated by wave 1, and tends to be a maximum near the equator. This partitioning depends upon the balance equation used. The different components of Φ have implications for how the observed Φ should and should not be used in diagnostic studies and provide a new interpretation as to what the ageostrophic component of the flow is. In particular, the ageostrophic wind is partitioned into the divergent wind plus contributions arising from the gradient of the PVAP and the gradient of the divergent geopotential rotated 90°.

The rotational and divergent geopotential fields and the PVAP have been computed from climatological mean January and July conditions. In addition for January, the tendencies due to the Coriolis form associated with the ageostrophic velocity are given and are shown to be related to the acceleration of jets in entrance regions and deceleration in exit regions of in excess of 30 m s−1/day for the Northern Hemisphere. However, while all three terms contributing to the ageostrophic velocity are of roughly equal importance overall, the divergent wind is less important in jet entrance and exit regions and the gradients of the PVAP and Φd terms are dominant. This illustrates the kinematic nature of these acceleration terms and shows the balance that exists in the momentum budget, but provides little insight into the cause of the existence of the mean jets.

## Abstract

It has been traditional in meteorology to divide the velocity field up into rotational and divergent components, but not the geopotential field. Yet any balance condition, such as the geostrophic relation or linear balance equation, is a diagnostic relation which states that not only can the balanced velocity field be computed from the geopotential field, but also that the geopotential can be derived from the velocity field. If the latter approach is adopted, then the difference between the observed and computed geopotential is the quantity we refer to as the divergent, or in some cases, ageostrophic, geopotential. In fact, for any balance set of equations, it is essential to partition the geopotential in such a way in order to derive an equivalent set of momentum equations.

The momentum equations equivalent to the linear balance set of equations are given. The linear balance equation is integrated to give a diagnostic relation between the rotational wind components and the gradient of the rotational geopotential plus an extra term which involves the gradient of the planetary vorticity advection potential (PVAP).

The partitioning of the velocity field into rotational and divergent parts v = vr + vd does not depend on any other field. Given vr and the associated streamfunction, we have computed Φ, from the linear balance equation. The difference Φd = Φ − Φr is of order Rossby number times Φ, has global scale dominated by wave 1, and tends to be a maximum near the equator. This partitioning depends upon the balance equation used. The different components of Φ have implications for how the observed Φ should and should not be used in diagnostic studies and provide a new interpretation as to what the ageostrophic component of the flow is. In particular, the ageostrophic wind is partitioned into the divergent wind plus contributions arising from the gradient of the PVAP and the gradient of the divergent geopotential rotated 90°.

The rotational and divergent geopotential fields and the PVAP have been computed from climatological mean January and July conditions. In addition for January, the tendencies due to the Coriolis form associated with the ageostrophic velocity are given and are shown to be related to the acceleration of jets in entrance regions and deceleration in exit regions of in excess of 30 m s−1/day for the Northern Hemisphere. However, while all three terms contributing to the ageostrophic velocity are of roughly equal importance overall, the divergent wind is less important in jet entrance and exit regions and the gradients of the PVAP and Φd terms are dominant. This illustrates the kinematic nature of these acceleration terms and shows the balance that exists in the momentum budget, but provides little insight into the cause of the existence of the mean jets.

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