A0
Zero-frequency wave–vortex coefficient array.
Description
Complex valued property with dimensions \((j,kl)\) and units of \(\mathrm{m^{2}\,s^{-1}}\).
Discussion
Zero-frequency wave–vortex coefficient array.
A0 stores the zero-frequency coefficients \(A_0^{k\ell j}\) with units of streamfunction, \(\mathrm{m^{2}\,s^{-1}}\). It is the active coefficient family for geostrophic and quasigeostrophic flow and, on transforms that include it, the mean-density anomaly.
The geostrophic solutions follow the decomposition in Early, Lelong, and Sundermeyer (2021). The complete basis, including the mean-density-anomaly solution, is described in the current available-potential-vorticity formulation.
For a three-dimensional transform with all four primary flow components, the constituents occupy A0 schematically as follows:
| Vertical mode | \(K_h=0\) | \(K_h>0\) |
|---|---|---|
| \(j=0\) | — | barotropic geostrophic |
| \(j>0\) | mean density anomaly | baroclinic geostrophic |
The transform’s mdaComponent.maskA0 and geostrophicComponent.maskA0 properties are the executable definitions of these regions. The table suppresses geometry-specific conjugate storage, excluded Nyquist modes, and antialiasing details. WVTransformStratifiedQG uses the applicable geostrophic A0 regions without a mean-density-anomaly component, while WVTransformBarotropicQG uses only nonzero-horizontal-wavenumber A0 modes.
For nonzero horizontal wavenumber, the geostrophic component mask identifies the active A0 locations. At zero horizontal wavenumber and internal vertical modes, the mean-density-anomaly mask identifies the horizontally averaged density anomaly relative to the no-motion state. These components are orthogonal in quadratic energy and potential enstrophy under the supported transform discretization.
The mean-density anomaly is deliberately retained: the horizontal-mean density in a domain need not equal the no-motion density profile used to define the transform. If those profiles are identical, the \(K_h=0,j>0\) coefficients vanish. The \(K_h=0,j=0\) location carries neither a geostrophic mode nor a common pressure-gauge degree of freedom.
A0 has no linear phase winding on the \(f\)-plane, so A0t equals A0. Quasigeostrophic transforms represent their supported state with A0 and have no active Ap or Am wave content.