## 1. Introduction

The potential vorticity (PV), especially the classic Rossby–Ertel PV (Rossby 1940; Ertel 1942), has been widely used for studies of atmospheric and oceanic dynamics (Hoskins et al. 1985; Robinson 1987; Davis and Emanuel 1991; Davis 1992; Bishop and Thorpe 1994; Stoelinga 1996; Thorpe and Volkert 1997; Lackmann 2002; Dong and Colucci 2005; Cordeira and Bosart 2010). When the hydrostatic PV (HPV) formulated in the height coordinate system is transformed into either the pressure or isentropic coordinate system, its form is simplified [as shown by (2.1)–(2.3) in this paper] and the associated conservation equation can be conveniently derived in a concise form. These HPV forms (in the pressure and isentropic coordinates) have been directly used in many HPV diagnostic studies (Huo et al. 1998; Davis and Bosart 2003; Gold and Nielsen-Gammon 2008; Gombos and Hansen 2008; Kieu and Zhang 2010). However, to apply these HPV forms to model-produced velocity and thermodynamic fields, the latter fields must be transformed from the model’s terrain-following coordinates to the pressure (or isentropic) coordinates. With this traditional approach, the transformed fields are subject to interpolation errors caused by the coordinate transformation. It is thus desirable to revisit the equivalent HPV form (Gerrity 1972) and derive its associated conservation equation in terrain-following coordinates, and then examine how to compute the HPV directly in the terrain-following coordinates to eliminate the interpolation error and thus improve the computational accuracy and efficiency. This is the motivation of this short contribution, and we have just learned from Dr. Mark Stoelinga (2011, personal communication) that he worked in this realm about 20 years ago and derived a similar conservation relationship (but not published) as presented in this contribution.

The paper is organized as follows. The next section reviews the equivalence of various HPV forms in height, pressure, isentropic, and terrain-following pressure coordinates and then derives the HPV conservation equation in the terrain-following pressure coordinate system. The derivation can be easily generalized for other HPV forms and their conservation equations in any vertical coordinates, and this is also explained in section 2. The HPV formulation derived in the terrain-following coordinate is used in section 3 to compute the HPV field from model-simulated wind and thermodynamic fields for a hurricane case, and the computed HPV field is used as a benchmark to evaluate the accuracies of the HPV fields computed (i) by the traditional approach (which applies the pressure-coordinate HPV formulation to interpolated fields on isobaric surfaces) and (ii) by using the Weather Research and Forecasting (WRF) model’s Read/Interpolate/Plot package (WRF RIP4; Stoelinga 2009). The results are summarized with conclusions in section 4.

## 2. HPV formulations and conservation property

### a. HPV formulations and their equivalence

*x*,

*y*,

*z*), the HPV has the following form:

*α*is the specific volume,

**∇**

_{3}≡ (

**∇**, ∂

*),*

_{z}**∇**≡ (∂

*, ∂*

_{x}*),*

_{y}**v**≡ (

*u*,

*υ*) is the horizontal velocity, and

*θ*is the potential temperature. In the pressure coordinate system, the HPV is transformed to

**∇**

*≡ (*

_{p}**∇**, ∂

*) and the hydrostatic equation*

_{p}*α*∂

*=*

_{z}p*−g*is used. Similarly, it is easy to verify that the HPV has the following equivalent form in the isentropic coordinate system:

**∇**

*≡ (*

_{θ}**∇**, ∂

*).*

_{θ}*x*,

*y*,

*η*) with

*η*≡ (

*p*−

*p*)/

_{t}*μ*and

*μ*=

*p*−

_{s}*p*, where

_{t}*p*is the surface pressure and

_{s}*p*is the pressure at the top boundary, the HPV in (2.2) is transformed into

_{t}**∇**

*≡ (*

_{η}**∇**, ∂

*). In the derivation of (2.4), the following chain rules are used:*

_{η}*s*represents

*x*,

*y*, or

*t*in the subscripts. The

*η*-coordinate HPV formulation in (2.4) recovers essentially the same result derived by Gerrity (1972). This HPV form is equivalent to those in (2.1)–(2.3) since they all represent the same HPV substance. Although this formulation is not new, its utilities and potential merits in HPV calculations have not been explored numerically. The conservation property retained by this form of HPV in the

*η*-vertical coordinate system is derived in the next subsection.

### b. RPV conservation equation

*f*plane in the

*η*-coordinate coordinate system are

*d*≡ ∂

_{t}*+*

_{t}**v**

_{3}·

**∇**

*,*

_{η}**v**

_{3}≡ (

**v**,

*ϖ*),

*ϖ*≡

*d*is the vertical velocity in

_{t}η*η*coordinates,

*α*=

*θ*(

*R*/

*p*)(

*p*/

*p*

_{0})

*,*

^{κ}*κ*=

*R*/

*c*,

_{p}*R*is the gas constant,

*c*is the specific heat under constant pressure, and =

_{p}*α*

**∇**

*p*|

*=*

_{z}*α*

**∇**

*p*|

*−*

_{η}*α*(∂

*)*

_{p}z**∇**

*z*|

*=*

_{η}*α*

**∇**

*p*|

*+*

_{η}**∇**

*ϕ*|

*is used [see Eqs. (1)–(44) of Haltiner and Williams (1980)].*

_{η}**∇**

*× (2.6a)” has the following form:*

_{η}**Z**≡

*f*

**k**+

**∇**

*×*

_{η}**v**is the absolute vorticity in

*η*coordinates, and (2.6c) is used in deriving the second last term. Note that

*α*is a function of

*p*and

*θ*as given above by

*α*=

*θ*(

*R*/

*p*)(

*p*/

*p*

_{0})

*. Using this function together with ∂*

^{κ}*=*

_{η}p*μ*, one can verify that

**∇**

*· [*

_{η}θ**k**×

**∇**(−

*μα*) +

**∇**

*× (*

_{η}*α*

**∇**

*p*)] =

*μJ*(

_{xy}*θ*,

*α*) + ∂

*(*

_{η}αJ_{xy}*p*,

*θ*) + ∂

*(*

_{η}θJ_{xy}*α*,

*p*) = 0, where

*J*( ) denotes the horizontal Jacobian differential operator on constant

_{xy}*η*surface. Thus, the dot product of

**∇**

*with (2.7) gives*

_{η}θ**Z**and

**∇**

*(2.6b) gives*

_{η}*η*-coordinate HPV, that is, PV

*(=−*

_{η}*g*

**Z**·

**∇**

*/*

_{η}θ*μ*), although PV

*is formulated in (2.4) by directly transforming the pressure-coordinate HPV formulation in (2.2) to the*

_{η}*η*-coordinate system without considering the original system of equations in (2.6). It is easy to see that the above proof can be extended to any other commonly used vertical coordinate. In this case, all the derivation steps in the above proof will remain the same except that

*η*is replaced by the new (arbitrary) coordinate

*σ*and

*μ*(=∂

*) is replaced by ∂*

_{η}p*. If*

_{σ}p*σ*is the hydrostatic pressure

*p*, then

*μ*is replaced by ∂

*= 1 and (2.10) reduces to the conventional pressure-coordinate HPV conservation equation. By using (2.4) and (2.10), the HPV can be directly computed and examined in the*

_{p}p*η*-coordinate system without transforming the model-produced fields to the pressure coordinate system. This utility is demonstrated by a numerical example in the next section.

## 3. Numerical example

In this section, (2.4) is used to calculate the HPV from the wind and thermodynamic fields of Hurricane Isabel at 1800 UTC (2 h after its landfall near Drum Inlet in North Carolina) on 18 September 2003 simulated by the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997). The nested COAMPS grid has a horizontal resolution of 6 km [see Fig. 9 of Zhao et al. (2008)] with 30 vertical levels on a terrain-following height coordinate system. For our purpose, the simulated wind and thermodynamic fields are interpolated from COAMPS’s terrain-following height coordinate to 30 *η* levels (with *p _{t}* = 50 mb). The interpolated data fields are treated as the original input fields (to mimic WRF-simulated fields in the

*η*-coordinate system) and can be used to compute the HPV directly (without interpolation) by using either (2.4) or WRF RIP4. The HPV field computed by the discretized (2.4) with the standard central finite differencing is still denoted PV

*. This PV*

_{η}*field is used as a benchmark to evaluate the error caused by the back and forth interpolations in the HPV field computed by the traditional approach. To mimic the traditional approach, the pressure-coordinate HPV formulation in (2.2) is discretized (with the standard central finite differencing) and applied to the vertically interpolated wind and thermodynamic fields on isobaric surfaces and then interpolated back to the original*

_{η}*η*surfaces. The final HPV field produced by this approach is still on the original

*η*grid and is still denoted PV

*.*

_{p}*η*surfaces in the WRF’s original coordinate system. By applying the chain rule in (2.5b) to the horizontal derivatives in (3.1), one can easily transform (3.1) into the pressure-coordinate HPV form in (2.2). Thus, the HPV in (3.1) is equivalent to that in (2.2) but is formulated in a hybrid manner in two coordinate systems. The original RIP had a version of PV calculation essentially identical to (2.4) for use with the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model’s (MM5’s)

*σ*-coordinate system, and this version was then elaborated into (3.1) around year 2000 when RIP was upgraded to be general enough that none of the diagnostic quantities was dependent on any specific vertical coordinate and all the model grids were treated as being on some native vertical coordinate not necessarily known for the calculations (M. Stoelinga 2011, personal communication). The HPV form in (3.1) has an advantage over the pressure-coordinate form in (2.2) because it computes the HPV directly from the WRF-simulated fields without vertical interpolation. However, the hydrostatic-pressure field needs to be stored and used to compute the vertical derivatives in (3.1). This procedure is unnecessary and can be eliminated, as revealed by the

*η*-coordinate HPV form in (2.4). Nevertheless, by assuming that the hydrostatic-pressure field is already available (or can be approximated by the total pressure), the use of (3.1) in RIP has an advantage in designing the code structure because it eliminates any need to even know what the vertical coordinate is or how it is mathematically defined for the input dataset.

The above three types of HPV fields are computed from the aforementioned wind and thermodynamic fields (on *η* surfaces). The computed HPV fields are plotted in the first three panels of Fig. 1 for PV* _{p}*, PV

*, and PV*

_{pη}*, respectively, at the vertical level of*

_{η}*η*= 0.84 over a 360 × 360 km

^{2}horizontal area cocentered with the hurricane eye. As shown, the PV

*field in Fig. 1b is almost identical to the PV*

_{pη}*field in Fig. 1c, and the difference is negligibly small and well within ±10*

_{η}^{−6}PV units (PVU) (not shown), where 1 PVU = 10

^{−6}m

^{2}s

^{−1}K kg

^{−1}. The PV

*field in Fig. 1a, however, is significantly different from the benchmark PV*

_{p}*field in Fig. 1c, and the difference goes beyond ±2 PVU around the hurricane eyewall and to the southwest outside the hurricane core, as shown in Fig. 1d. The difference field in Fig. 1d reveals that the errors caused by the interpolations in computing HPV with the traditional approach can be quite large in areas where the HPV and/or HPV gradient are large. Similar differences are seen on other*

_{η}*η*levels, although the difference diminishes as

*η*approaches 0 (the top level where the

*η*coordinate becomes locally the same as the pressure coordinate).

For further and fair comparisons, the above PV* _{pη}* and PV

*fields are interpolated vertically onto the same isobaric surfaces as those used by the aforementioned traditional approach. In this case, the PV*

_{η}*field computed by the traditional approach can be compared directly with the interpolated PV*

_{p}*field on each isobaric surface without interpolating back onto the*

_{η}*η*surfaces. As shown by the HPV fields plotted at the isobaric level of

*p*= 870 mb in the first three panels of Fig. 2, the PV

*field in Fig. 2b is still nearly identical to the PV*

_{pη}*field in Fig. 2c, and the difference is still well within ±10*

_{η}^{−6}PVU (not shown). The PV

*field in Fig. 2a is still significantly different from the benchmark PV*

_{p}*field in Fig. 2c. In particular, as shown in Fig. 2d, the difference remains very large (±2 PVU) around the hurricane eyewall, especially along the southern rim, although the difference (on*

_{η}*p*= 870 mb) is not as large as that (on

*η*= 0.84) in Fig. 1d in the area to the southwest outside the hurricane core. The vertical structures of the above computed PV

*and PV*

_{p}*fields are shown in the vertical cross sections plotted (in the pressure coordinates at*

_{η}*x*= 180 km through the hurricane center) in Figs. 3a and 3b, respectively. As shown, PV

*has roughly the same vertical structure as PV*

_{p}*but with much reduced intensity, especially in the lower middle troposphere around the hurricane center.*

_{η}As in Fig. 1, but for fields computed at the constant pressure level of *p* = 870 mb.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

As in Fig. 1, but for fields computed at the constant pressure level of *p* = 870 mb.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

As in Fig. 1, but for fields computed at the constant pressure level of *p* = 870 mb.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

Vertical cross sections of (a) PV* _{p}* and (b) PV

*at*

_{η}*x*= 180 km through the hurricane center. The contours are labeled in PVU.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

Vertical cross sections of (a) PV* _{p}* and (b) PV

*at*

_{η}*x*= 180 km through the hurricane center. The contours are labeled in PVU.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

Vertical cross sections of (a) PV* _{p}* and (b) PV

*at*

_{η}*x*= 180 km through the hurricane center. The contours are labeled in PVU.

Citation: Monthly Weather Review 139, 9; 10.1175/MWR-D-11-00083.1

## 4. Conclusions

In this paper, the hydrostatic potential vorticity (HPV) formulated in terrain-following coordinates (Gerrity 1972) is revisited with its conservation equation derived from the original system of primitive equations. In particular, the HPV formulated in a terrain-following pressure coordinate system, such as the one used in the WRF model (Skamarock et al. 2008), is shown to be equivalent to the widely used HPV formulations in height, pressure, and isentropic coordinates in the sense that they all represent the same HPV substance and retain the same conservation property. Similar equivalent HPV formulations (not shown in this paper) can be derived in other terrain-following coordinate systems, including the terrain-following height coordinate system used in COAMPS (Hodur 1997) and the Advanced Regional Prediction System (ARPS; Xue et al. 2001). When such a HPV formulation is derived for a given model’s terrain-following coordinate system, it can be applied directly to model-simulated velocity and thermodynamic fields on the model’s original terrain-following grid without coordinate transformation and thus eliminate grid interpolation error. This is an apparent advantage for the equivalent HPV formulation in model’s terrain-following coordinates, but such an advantage was not explored numerically or utilized in HPV computation in the literature, as briefly reviewed in the introduction.

To explore the above advantage and evaluate its significance, three types of HPV fields, denoted PV* _{p}*, PV

*, and PV*

_{pη}*, respectively, are computed from simulated wind and thermodynamic fields (on terrain-following pressure coordinates) for a hurricane case (Zhao et al. 2008). The PV*

_{η}*field is computed, as in the traditional approach, by applying the pressure-coordinate HPV formulation [see (2.2)] to the wind and thermodynamic fields interpolated on isobaric surfaces interpolated back to the original*

_{p}*η*surfaces. The PV

*field is computed by the hybrid HPV formulation [see (3.1)] in WRF RIP4 (Stoelinga 2009). The PV*

_{pη}*field is computed by the HPV formulation in the terrain-following pressure coordinates [see (2.4)]. Although these three HPV fields represent the very same HPV substance and thus should be identical, only PV*

_{η}*and PV*

_{pη}*are found to be nearly identical to each other. The difference between PV*

_{η}*and PV*

_{p}*is found to be very significant, and it can reach 33% of the amplitude (maximum) of the benchmark PV*

_{η}*field in the lower middle-tropospheric area around the hurricane center (as shown in Figs. 1 and 2). In the very same area, the HPV and/or HPV gradient are large and the terrain-following coordinate surfaces are severely tilted and/or deformed (with large curvatures) away from their counterpart constant pressure surfaces, so larger interpolation errors are caused when the simulated wind and thermodynamic fields are transformed into the pressure coordinates to compute PV*

_{η}*and/or the compute PV*

_{p}*is transformed back to the model’s original coordinates. This explains the large difference between PV*

_{p}*and PV*

_{p}*in the above area, and the difference is caused by the interpolation error in the computation of PV*

_{η}*. It is thus more efficient but also more or much more accurate to compute PV*

_{p}*than PV*

_{η}*from model-simulated fields.*

_{p}For the above hurricane case, the low surface pressure around the hurricane center acts like an isolated “mountain” in the pressure coordinate system. This mountain is relatively smooth although its “topography” is enhanced slightly by the coast terrain underneath the landfall hurricane Isabel (to the east of the Appalachian Mountains). Real mountainous topography usually contains much more intense small-scale structures than the surface pressure field for the hurricane case examined in this paper. When HPV is computed from high-resolution model-simulated wind and thermodynamic fields over a major mountain range (such as the Rocky Mountains), the intense small-scale terrain structures may constitute a much more challenging test for the accuracy of HPV diagnostic computations. Such a test will be performed in our subsequent HPV diagnostic studies beyond this short contribution.

Finally, the HPV conservation formulated in the terrain-following coordinates [see (2.10)] can be used together with the PV* _{η}* formulation (or the RIP PV

*formulation) to facilitate both diagnostic and prognostic studies of model-simulated HPV dynamics. These utilities and their merits deserve continued studies, although the HPV obtained from model-simulated native datasets will not strictly follow the conservation equation unless the model itself is hydrostatic and the simulated process is nondissipative and adiabatic.*

_{pη}## Acknowledgments

We are thankful to Dr. Mark Stoelinga and the anonymous reviewer for their comments and suggestions that improved the quality and accuracy of the presentation of the paper. Dr. Mark Stoelinga generously shared his insights on HPV computations and provided accurate information on the history of RIP development (cited in the paper). The COAPMS simulated data fields used in this study are provided graciously by Dr. Qingyun Zhao and Li Wei. The research was supported by the ONR Grant N000141010778 to the University of Oklahoma, and by China’s National Natural Sciences Foundation Grants 40930950 and 40775031 to the Institute of Atmospheric Physics. Funding was also provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227, U.S. Department of Commerce.

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