• Bracegirdle, T. J., and S. L. Gray, 2009: The dynamics of a polar low assessed using potential vorticity inversion. Quart. J. Roy. Meteor. Soc., 135, 880893, doi:10.1002/qj.411.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Charney, J., 1955: The use of primitive equations of motion in numerical prediction. Tellus, 7, 2226, doi:10.1111/j.2153-3490.1955.tb01138.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, C. A., 1992: Piecewise potential vorticity inversion. J. Atmos. Sci., 49, 13971411, doi:10.1175/1520-0469(1992)049<1397:PPVI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 19291953, doi:10.1175/1520-0493(1991)119<1929:PVDOC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553587, doi:10.1002/qj.828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Egger, J., and K.-P. Hoinka, 2010: Potential temperature and potential vorticity inversion: Complementary approaches. J. Atmos. Sci., 67, 40014016, doi:10.1175/2010JAS3532.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ertel, H., 1942: Ein neuer hydrodynamischer Wirbelsatz. Meteor. Z., 59, 277281.

  • Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of isentropic distributions of potential vorticity in the presence of diabatic heating and fictional or other forces. J. Atmos. Sci., 44, 828841, doi:10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946, doi:10.1002/qj.49711147002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., 2015: Potential vorticity. Encyclopedia of Atmospheric Sciences, 2nd ed. G. R. North, J. Pyle, and F. Zhang, Eds., Vol. 2, Elsevier, 375–383, doi:10.1016/B978-0-12-382225-3.00140-7.

    • Crossref
    • Export Citation
  • Schär, C., M. Sprenger, D. Lüthi, Q. Jiany, R. Smith, and R. Benoit, 2003: Structure and dynamics of an alpine potential-vorticity banner. Quart. J. Roy. Meteor. Soc., 129, 825855, doi:10.1256/qj.02.47.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, P., 1961: Numerical Weather Analysis and Prediction. MacMillan, 170 pp.

  • Thorpe, A., 1985: Diagnosis of balanced vortex structures using potential vorticity. J. Atmos. Sci., 42, 397406, doi:10.1175/1520-0469(1985)042<0397:DOBVSU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vallis, G., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 745 pp.

    • Crossref
    • Export Citation
  • Viúdez, Á., 2001: The relation between Beltrami’s material vorticity and Rossby Ertel’s potential vorticity. J. Atmos. Sci., 58, 25092517, doi:10.1175/1520-0469(2001)058<2509:TRBBMV>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Viúdez, Á., 2012: Potential vorticity and inertia–gravity waves. Geophys. Astrophys. Fluid Dyn., 106, 6788, doi:10.1080/03091929.2010.537265.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zdunkowski, W., and A. Bott, 2003: Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 719 pp.

    • Crossref
    • Export Citation
  • View in gallery

    Orientation of the various basis and vorticity vectors with respect to an η surface.

  • View in gallery

    Vorticity and Montgomery potential on the θ = 285-K surface at 0000 UTC 12 Feb 2008: (a) vorticity (10−4 s−1; contour interval = 0.5 × 10−4 s−1) and (b) Montgomery potential on θ = 285 K (103 m2 s−2; contour interval = 1.0 × 103 m2 s−2). Negative values and areas outside the intersection contour are shaded. Mean value subtracted in (b).

  • View in gallery

    Results of inversion. Montgomery potential on the θ = 285-K surface at 0000 UTC 12 Feb 2008 (103 m2 s−2) (a) as obtained by inverting as in Fig. 2a, (b) as obtained by inverting , and (c) the difference of (a) and (b). The contour interval is 0.5 × 103 m2 s−2. Negative values and areas outside the intersection contour are shaded. Mean value subtracted in (a) and (b).

  • View in gallery

    Piecewise inversion of on θ = 285-K surface at 0000 UTC 12 Feb 2008. Montgomery potential (102 m2 s−2) for except in the sector (a) 90° < λ < 120°E and (b) 120° < λ < 150°E. The contour interval is 0.2 × 102 m2 s−2. Negative values and areas outside the intersection contour are shaded.

  • View in gallery

    PVDI for the 500-hPa surface: (a) observed geopotential ϕ at 0000 UTC 12 Feb 2008 (102 m2 s−2); (b) inverted ϕ field for observed ; and (c) as in (b), but for . The contour interval is 1.0 × 102 m2 s−2. Negative values and areas outside the intersection contour are shaded. Subtracted mean values of : (a) 0.439 × 105, (b) 0.440 × 105, and (c) 0.444 × 105 m2 s−2.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 93 93 19
PDF Downloads 16 16 2

Inversion of Potential Vorticity Density

View More View Less
  • 1 Meteorological Institute, University of Munich, Munich, Germany
  • 2 Institute for Atmospheric Physics, Deutsches Zentrum für Luft- und Raumfahrt, Oberpfaffenhofen, Germany
  • 3 Geophysical Institute, University of Bergen, Bergen, Norway
© Get Permissions
Full access

Abstract

Inversion of potential vorticity density with absolute vorticity and function η is explored in η coordinates. This density is shown to be the component of absolute vorticity associated with the vertical vector of the covariant basis of η coordinates. This implies that inversion of in η coordinates is a two-dimensional problem in hydrostatic flow.

Examples of inversions are presented for (θ is potential temperature) and (p is pressure) with satisfactory results for domains covering the North Pole. The role of the boundary conditions is investigated and piecewise inversions are performed as well. The results shed new light on the interpretation of potential vorticity inversions.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Joseph Egger, j.egger@lrz.uni-muenchen.de

Abstract

Inversion of potential vorticity density with absolute vorticity and function η is explored in η coordinates. This density is shown to be the component of absolute vorticity associated with the vertical vector of the covariant basis of η coordinates. This implies that inversion of in η coordinates is a two-dimensional problem in hydrostatic flow.

Examples of inversions are presented for (θ is potential temperature) and (p is pressure) with satisfactory results for domains covering the North Pole. The role of the boundary conditions is investigated and piecewise inversions are performed as well. The results shed new light on the interpretation of potential vorticity inversions.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Joseph Egger, j.egger@lrz.uni-muenchen.de

1. Introduction

Potential vorticity (PV) is an important variable in dynamic meteorology and oceanography and is widely used for the simulation and interpretation of a broad range of flow phenomena (e.g., Vallis 2006), where potential vorticity is
e1
and is absolute vorticity, η is a function of space and time, and ρ is density (Ertel 1942). Use of is not widespread except for (θ is potential temperature) where is conserved in adiabatic and inviscid flow. The explicit expression for in spherical coordinates is fairly complicated:
e2
with standard notation (longitude λ; latitude φ; height z; velocity components u, υ, and w; Coriolis parameter with ; and Earth’s radius a). The traditional approximation is accepted in (2), where we assume r = a + za and neglect Coriolis terms with (e.g., Vallis 2006). A simplification can be obtained by selecting η as a vertical coordinate and turning to PV density (PVD) . We have to realize, however, that is a density in (λ, φ, z) space, but not in (λ, φ, η) space, as would be appropriate in η coordinates. We introduce the density to ensure that volume integrals of in height coordinates equal those of in η coordinates.
The transformation of (2) to η coordinates can be performed by introducing the covariant basis vectors
e3
where are the standard spherical unit vectors with pointing eastward, pointing northward, and pointing upward [e.g., Zdunkowski and Bott (2003), see their Fig. 1]. The first two vectors are embedded in η surfaces and orthogonal to with contravariant basis vector , where . Next we have to adapt the derivatives in (2) to the η system so that the absolute vorticity, after expressing all in terms of , becomes
e4
with
e5
where all “horizontal” derivatives are performed for constant η and is the height of η surfaces. The final step consists in multiplying (4) by , so that
e6
On the other hand, results if we multiply (4) by . An alternative derivation of (6) is provided by Viúdez (2001).

Because in (4) is a unit vector and because we may replace and in (4) by unit vectors multiplied by , we see that is the vertical component of absolute vorticity with respect to a nonorthogonal basis of unit vectors parallel to the covariant ones. In particular, is the related vertical component of relative vorticity in hydrostatic flow where in (4)(6). This interpretation differs somewhat from others found in the literature. For example, McIntyre (2015) claims that “the isentropic vorticity… is the same as the component of the vorticity vector normal to the isentropic surface” (p. 376). This definition converges to ours for steepness (see Fig. 1). Small values of 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 . This formula is well known for isentropic flow.

Fig. 1.
Fig. 1.

Orientation of the various basis and vorticity vectors with respect to an η surface.

Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0133.1

With available we can now turn to inversion. PV inversion (PVI) is one of the most popular applications of PV thinking (Hoskins et al. 1985), which is used to derive winds, pressure, and temperature from a field on the basis of a balance condition and suitable boundary conditions (e.g., Thorpe 1985; Hoskins et al. 1985). However, inversion of is difficult owing to the nonlinearity of so that iterative methods have to be used (e.g., Davis 1992). This problem can be partly overcome by recognizing that invertibility is not restricted to PV (Egger and Hoinka 2010). In particular, we may perform an inversion of (PVDI) in η coordinates. For example, is a linear two-dimensional expression on isentropic surfaces [see (6)], so that inversion is relatively simple.

Although is materially conserved for adiabatic and inviscid flow, all other choices of PV like are not conserved nor are and . Thus, can be stepped forward with the winds obtained from the inversion, while that is not possible for and and . However, as will be discussed in detail below, inversion of allows to evaluate so that the inversions of and are equivalent with respect to eventual predictions. Moreover, PVDI is of interest by itself, because it attributes the flow on an η surface to the vorticity on that surface, at least for geostrophic balance, as will be shown below. PVI would attribute the flow to the three-dimensional field . Thus, the same flow can be attributed to different “sources” depending on the variable selected for inversion.

It is the purpose of this short contribution to present inversions of and also of based on observations to demonstrate the feasibility of this approach and to discuss the interpretation of inversions.

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 in pressure coordinates.

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 for and , where the main step involves the derivation of the flow on an η surface from observed on that surface. Piecewise inversions will be carried out as well.

In general, a streamfunction ψ can be obtained by inverting , with two-dimensional Laplacian . This is a linear problem. Geostrophic balance, or a more advanced balance condition like that of Charney (1955), must then be used to obtain, for example, the Montgomery potential for or the geopotential ϕ for . Although the latter condition is nonlinear with respect to ψ, it is linear with respect to M or ϕ.

a. Inversion of

We select the distribution of on the surface 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.

Fig. 2.
Fig. 2.

Vorticity and Montgomery potential on the θ = 285-K surface at 0000 UTC 12 Feb 2008: (a) vorticity (10−4 s−1; contour interval = 0.5 × 10−4 s−1) and (b) Montgomery potential on θ = 285 K (103 m2 s−2; contour interval = 1.0 × 103 m2 s−2). Negative values and areas outside the intersection contour are shaded. Mean value subtracted in (b).

Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0133.1

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 on this surface, as determined from ERA-Interim (Dee et al. 2011), is fairly patchy, but there are several stripes of positive as well as negative vorticity extending from the southern boundary almost to the pole (Fig. 2a). The observed 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 , as expected.

Accepting geostrophic balance with geostrophic winds
e7
e8
inversion of requires to solve
e9
Observed values 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 yields the M patterns in Figs. 3 and 4, where the area-mean has been subtracted. The inverted 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.

Fig. 3.
Fig. 3.

Results of inversion. Montgomery potential on the θ = 285-K surface at 0000 UTC 12 Feb 2008 (103 m2 s−2) (a) as obtained by inverting as in Fig. 2a, (b) as obtained by inverting , and (c) the difference of (a) and (b). The contour interval is 0.5 × 103 m2 s−2. Negative values and areas outside the intersection contour are shaded. Mean value subtracted in (a) and (b).

Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0133.1

Fig. 4.
Fig. 4.

Piecewise inversion of on θ = 285-K surface at 0000 UTC 12 Feb 2008. Montgomery potential (102 m2 s−2) for except in the sector (a) 90° < λ < 120°E and (b) 120° < λ < 150°E. The contour interval is 0.2 × 102 m2 s−2. Negative values and areas outside the intersection contour are shaded.

Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0133.1

The role of the prescribed boundary values can be explored by inverting a vanishing relative vorticity (Fig. 3b) but keeping the same boundary values as in Fig. 3a. This inversion of boundary values is inspired by the standard practice in PVI to determine the impact of boundary values on distant flows (e.g., Davis and Emanuel 1991). This technique is partly motivated by the idea that potential temperature at the lower boundary can be interpreted as a PV anomaly that exerts an impact on the flow (Hoskins et al. 1985). Although this interpretation cannot be extended to our case, the boundary values of M indicate direction and intensity of the geostrophic flow across the boundary. Thus, inversions with tell us how these fluxes can be maintained by a flow in the interior without vorticity.

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 away from the boundary at . Thus, the smallest wavenumbers dominate the far field yielding a fairly smooth pattern away from the boundary.

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 and amplitudes are generally small in the difference pattern in Fig. 3c. Note the reduced contour interval in Fig. 3c. We attribute this difference to the vorticity anomalies on the 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 can be used to determine the pressure, provided the surface temperature is known. With that, even would be available and could be predicted using the available geostrophic winds.

Piecewise inversion has to select features of the vorticity field in Fig. 2a. Inversion is then performed with at the boundaries. For example, the sector 90° < 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 at the boundaries, outside the domain 90°–120°E, and observed vorticity inside. The solution is centered in the longitude sector with a high in the south and a small low in the north, though the high extends into the adjacent sectors.

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 are relatively rare, though Haynes and McIntyre (1987) discussed fluxes of . Note that hydrostatic PVD and PV are the same for except for a factor . Thus, PVI is the same as PVDI. The equation to be solved is (9), where we have to replace M by the geopotential ϕ and by . The boundary conditions are the values of the geopotential at the boundary. We chose the 500-hPa surface, which rarely intersects the ground. The selected boundary contour is the same as before, which is an unusual choice for a pressure surface but was chosen to aid the comparison with the previous case. Such inversions have a long tradition and have been carried out routinely in the early one-layer models of numerical forecasting (Thompson 1961). It is nevertheless of interest to perform an inversion of in parallel to .

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 (Fig. 5c) yields a pattern that captures much of Fig. 5a and documents the importance of the lateral boundary values. As before, these boundary values of ϕ determine the geostrophic flows across the boundary. Areal mean values have been subtracted for the respective fields. Note that the Arctic low has no closed height line in Fig. 5b, as is required for flows without vorticity. As the height of the 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, is readily available on this isobaric surface owing to the simple relationship with .

Fig. 5.
Fig. 5.

PVDI for the 500-hPa surface: (a) observed geopotential ϕ at 0000 UTC 12 Feb 2008 (102 m2 s−2); (b) inverted ϕ field for observed ; and (c) as in (b), but for . The contour interval is 1.0 × 102 m2 s−2. Negative values and areas outside the intersection contour are shaded. Subtracted mean values of : (a) 0.439 × 105, (b) 0.440 × 105, and (c) 0.444 × 105 m2 s−2.

Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0133.1

3. Concluding remarks

This study has been stimulated by the well-known result that isentropic hydrostatic PVD reduces to a vorticity in isentropic coordinates. The variable η, as specified in the definition of , has been chosen as a vertical coordinate in extension of the isentropic case and the PVD is considered instead of . The vorticity turns out to be the absolute vertical vorticity component with respect to the basis of unit vectors aligned with the covariant basis. The nonhydrostatic terms of can be important for strongly nonhydrostatic flows, such as in PV banners.

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 in η coordinates, however, led us to consider the inversion of on η surfaces, which reduces to a two-dimensional linear problem for hydrostatic flow. Such inversions have been carried out for and . Geostrophic balance yielded satisfactory results in both cases. The role of the boundary values has been investigated by conducting inversions with in the domain. It turned out that a substantial part of the observed 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 on an η surface is a “source” for the flow on that surface, but boundary values are also important. On the other hand, inversion of would result in different attributions.

The simplicity of the hydrostatic inversion in η coordinates is lost if we turn to nonhydrostatic flows. The contribution of w to is difficult to evaluate, because the inversion becomes inherently nonlinear and three-dimensional [see Viúdez (2012) for nonhydrostatic inversions in a Boussinesq fluid].

Acknowledgments

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.

REFERENCES

  • Bracegirdle, T. J., and S. L. Gray, 2009: The dynamics of a polar low assessed using potential vorticity inversion. Quart. J. Roy. Meteor. Soc., 135, 880893, doi:10.1002/qj.411.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Charney, J., 1955: The use of primitive equations of motion in numerical prediction. Tellus, 7, 2226, doi:10.1111/j.2153-3490.1955.tb01138.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, C. A., 1992: Piecewise potential vorticity inversion. J. Atmos. Sci., 49, 13971411, doi:10.1175/1520-0469(1992)049<1397:PPVI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 19291953, doi:10.1175/1520-0493(1991)119<1929:PVDOC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553587, doi:10.1002/qj.828.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Egger, J., and K.-P. Hoinka, 2010: Potential temperature and potential vorticity inversion: Complementary approaches. J. Atmos. Sci., 67, 40014016, doi:10.1175/2010JAS3532.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ertel, H., 1942: Ein neuer hydrodynamischer Wirbelsatz. Meteor. Z., 59, 277281.

  • Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of isentropic distributions of potential vorticity in the presence of diabatic heating and fictional or other forces. J. Atmos. Sci., 44, 828841, doi:10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946, doi:10.1002/qj.49711147002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., 2015: Potential vorticity. Encyclopedia of Atmospheric Sciences, 2nd ed. G. R. North, J. Pyle, and F. Zhang, Eds., Vol. 2, Elsevier, 375–383, doi:10.1016/B978-0-12-382225-3.00140-7.

    • Crossref
    • Export Citation
  • Schär, C., M. Sprenger, D. Lüthi, Q. Jiany, R. Smith, and R. Benoit, 2003: Structure and dynamics of an alpine potential-vorticity banner. Quart. J. Roy. Meteor. Soc., 129, 825855, doi:10.1256/qj.02.47.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thompson, P., 1961: Numerical Weather Analysis and Prediction. MacMillan, 170 pp.

  • Thorpe, A., 1985: Diagnosis of balanced vortex structures using potential vorticity. J. Atmos. Sci., 42, 397406, doi:10.1175/1520-0469(1985)042<0397:DOBVSU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vallis, G., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 745 pp.

    • Crossref
    • Export Citation
  • Viúdez, Á., 2001: The relation between Beltrami’s material vorticity and Rossby Ertel’s potential vorticity. J. Atmos. Sci., 58, 25092517, doi:10.1175/1520-0469(2001)058<2509:TRBBMV>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Viúdez, Á., 2012: Potential vorticity and inertia–gravity waves. Geophys. Astrophys. Fluid Dyn., 106, 6788, doi:10.1080/03091929.2010.537265.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zdunkowski, W., and A. Bott, 2003: Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 719 pp.

    • Crossref
    • Export Citation
Save