We investigate a perturbation form of the vertical momentum equation using output from the Met Office's Unified Model at high resolution (grid-length 1.5 km). The terms of the perturbation equation are grouped into five different types (listed below) and evaluated with model data. The spatial structures are plotted at a number of model levels and Fourier transformed to give the contribution of each term as a function of scale.
Our aim is to assess the vertical momentum budget at high resolution. This is important, for instance, in variational data assimilation (VAR). In standard VAR systems that work at synoptic scales (such as that of the Met Office) background errors between temperature and pressure are assumed to be related via the hydrostatic relation. Scale analysis shows that the hydrostatic equation is no longer expected to be relevant at the convective scale, and so we look at the remaining parts of the vertical momentum budget. This is preliminary work with the eventual aim of building a possible new diagnostic relationship between temperature and pressure errors which is valid at the convective scale.
The basic vertical momentum equation between full (unperturbed) variables is
where 
is density, 
, 
, and 
are the three components of velocity, 
is the acceleration due to gravity, 
, 
, and 
are the three spatial dimensions and 
is time. Define the following perturbations
where the '0' subscript represents a reference state obeying (1) and primed variables are perturbations. The equation that is obeyed by the perturbations is found by substituting (2) into (1). This gives a large number of terms which we group according to five types: (a) linear tendency terms (b) non-linear tendency terms, (c) linear advection terms, (d) non-linear advection terms and (e) residual non-hydrostatic terms. Linear (non-linear) terms are defined as those that contain (more than) one perturbation variable.
Terms (3a) to (3e) are plotted on a number of levels and the terms are Fourier transformed to give the contribution to each horizontal scale. If one of (3a) to (3e) is 
, and its Fourier transform is 
(a complex number), then define the weight
where the sum is over 
which lie in a thin annulus of radius 
in spectral space
The parameter 
is the total wavenumber. Weight 
may also be plotted as a function of lengthscale, 
where 
.
Due to multi-scale interactions between perturbations, and between perturbations and the reference state, the function 
or 
cannot be interpreted as the contribution of a given term type due to flow at scale 
or 
. Instead it should be interpreted as the contribution of the combination of the terms that make up each term at scale 
or 
. This is a limitation of this approach. The only exception is the non-hydrostatic residual (3e), which appears as a linear term with no reference state multiplier.