## 1. Introduction

*η*is a function of space and time, and

*ρ*is density (Ertel 1942). Use of

*θ*is potential temperature) where

*λ*; latitude

*φ*; height

*z*; velocity components

*u*,

*υ*, and

*w*; Coriolis parameter

*a*). The traditional approximation is accepted in (2), where we assume

*r*=

*a*+

*z*≈

*a*and neglect Coriolis terms with

*η*as a vertical coordinate and turning to PV density (PVD)

*λ*,

*φ*,

*z*) space, but not in (

*λ*,

*φ*,

*η*) space, as would be appropriate in

*η*coordinates. We introduce the density

*η*coordinates.

*η*coordinates can be performed by introducing the covariant basis vectorswhere

*η*surfaces and orthogonal to

*η*system so that the absolute vorticity, after expressing all

*η*and

*η*surfaces. The final step consists in multiplying (4) by

Because *α* are typical of isentropic surfaces in large-scale flow, but the vorticity must be formulated with respect to the nonorthogonal vector basis to be correct also for steep surfaces as found, for example, in PV banners (Schär et al. 2003). In what follows we will concentrate on hydrostatic flows so that

With *η* coordinates. For example,

Although *η* surface to the vorticity

It is the purpose of this short contribution to present inversions of

## 2. Inversion of potential vorticity density

As stated above, PVI is a well explored and widely used technique to approximately capture all dynamic information about a flow state (e.g., Thorpe 1985). Only PV has to be known, if a balance relation is imposed together with appropriate boundary conditions. PVI has mainly been carried out for

Piecewise potential vorticity inversion (PPVI) goes one step further by seeking to determine the flow fields associated with isolated PV anomalies. This technique has been used to understand, for example, the impact of observed PV anomalies on hurricane development (Davis and Emanuel 1991) or the influence of upper-level PV features on the evolution of polar lows (Bracegirdle and Gray 2009). We wish to invert *η* surface from observed

In general, a streamfunction *ψ* can be obtained by inverting *ϕ* for *ψ*, it is linear with respect to *M* or *ϕ*.

### a. Inversion of

We select the distribution of *θ* = 285 K in the Northern Hemisphere for a demonstration of PVDI (see Fig. 2). The date in Fig. 2 has been chosen randomly, as we do not aim to perform a dynamic analysis of a certain flow configuration. The main purpose of this presentation is to discuss PVDI as a method.

The *θ* = 285-K surface intersects Earth’s surface all around the North Pole on that day and forms a dome north of the intersection contour. The observed vorticity *M* perturbations are dominated by a huge ridge covering much of western Eurasia and a system of lows closer to the pole (Fig. 2b), where a northward decrease of *M* implies westerly flow. The scale of the observed *M* perturbations, defined as the deviation from the areal mean, is much larger than that of

*M*are prescribed where isentropic surfaces intersect the ground.

A circular domain of radius 450 km covering the North Pole is excluded from the inversion to avoid technical problems due to convergence of the meridians. Observed values of *M* are prescribed at this bounding circle. Relaxation with a convergence threshold of *M* patterns in Figs. 3 and 4, where the area-mean *M* field (Fig. 3a) satisfactorily approximates the observations in Fig. 2b. The inversion turns the complicated vorticity distribution in Fig. 2a into a relatively simple *M* pattern.

The role of the prescribed boundary values can be explored by inverting a vanishing relative vorticity *M* indicate direction and intensity of the geostrophic flow across the boundary. Thus, inversions with

A gross estimate of the response to boundary values can be based on *f*-plane solutions. In these cases, boundary perturbations of wavenumber *k* at a zonal boundary with meridional coordinate *y* decay proportional to

The pattern in Fig. 3b is indeed quite smooth and the *M* values at the boundary extend far into the domain. Figures 3a and 3b are quite similar with positive values over Central Asia and a large depression extending from the Pacific across the North Pole as in Fig. 3a. In other words, the role of the boundary values in the inversion is at least as important as that of *θ* surfaces.

The height of the isentropic surface in Fig. 3a cannot be derived from the inverted *M* values on just one isentropic surface. We would have to solve (9) on a stack of *θ* surfaces so that the hydrostatic relation

Piecewise inversion has to select features of the vorticity field in Fig. 2a. Inversion is then performed with *λ* < 120°E contains patches of negative relative vorticity, say, south of 70°N and a positive anomaly close to the North Pole. Figure 4a shows the Montgomery potential obtained with

There are patches of strong positive vorticity in the sector 120°–150° (Fig. 2a), which correspond to the eastward-extending trough (Fig. 2b). The PPVDI results for this sector have also a low near the pole, which corresponds to a vorticity maximum there (see Fig. 4b).

Amplitudes in Fig. 4 are smaller but of the same order of magnitude as in Fig. 2b. That is to be expected, because the impact of the boundary values is missing in Fig. 4. Piecewise inversion could be performed for all latitude sectors, where superposition of all their results would give Fig. 3c. Comparison of Figs. 4a,b to Fig. 3c leads to the conclusion that the vorticity in one sector almost completely determines *M* in that sector in Fig. 3c.

### b. Inversion of

Investigations of *M* by the geopotential *ϕ* and

The observed *ϕ* field in Fig. 5a is similar to the *M* pattern in Fig. 3a with an Asian ridge and lows at the North Pole, over North America, and over the Pacific. The inversion is again satisfactory (Fig. 5b) with a distinct Arctic low. The inversion for *ϕ* determine the geostrophic flows across the boundary. Areal mean values *p* surfaces is given by the geopotential, PVDI in the isobaric case yields the complete information and we do not have to solve (9) for a stack of isobaric surfaces. Moreover,

## 3. Concluding remarks

This study has been stimulated by the well-known result that isentropic hydrostatic PVD *η*, as specified in the definition of

Three-dimensional PVI requires iterative methods to reconstruct the complete flow from the PV field in order to associate flow features with PV anomalies. The relatively simple structure of *η* coordinates, however, led us to consider the inversion of *η* surfaces, which reduces to a two-dimensional linear problem for hydrostatic flow. Such inversions have been carried out for *M* (*ϕ*) field is related to the conditions at the boundaries, which represent the geostrophic wind across the boundaries and, thus, the dynamic interaction with the surrounding atmosphere.

We also conducted piecewise inversion to explore the role of isolated PV features. Examples of PPVDI have been presented, where we evaluated the geostrophic streamfunction associated with the vorticity in various longitude sectors and their extension into neighboring sectors.

Attribution appears to be straightforward in our case. The vorticity *η* surface is a “source” for the flow on that surface, but boundary values are also important. On the other hand, inversion of

The simplicity of the hydrostatic *η* coordinates is lost if we turn to nonhydrostatic flows. The contribution of *w* to

We are grateful to two reviewers whose detailed comments helped to improve the paper. We would like to acknowledge the use of the ERA-Interim data produced and provided by ECMWF.

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