Advances in Meteorology

Volume 2016 (2016), Article ID 2908423, 11 pages

http://dx.doi.org/10.1155/2016/2908423

## The Application of Diabatic Heating in -Vectors for the Study of a North American Cyclone Event

^{1}Department of Soil, Environmental, and Atmospheric Science, University of Missouri, 302 E ABNR Building, Columbia, MO 65211, USA^{2}National Oceanic and Atmospheric Administration, National Weather Service, Operations Proving Ground, 7220 NW 101st Terrace, Kansas City, MO 64153, USA^{3}National Weather Service Portland Weather Forecast Office, 5241 NE 122nd Avenue, Portland, OR 97230-1089, USA^{4}The Weather Channel, 300 Interstate North Parkway, Atlanta, GA 30339, USA

Received 14 November 2014; Revised 7 May 2015; Accepted 25 May 2015

Academic Editor: Klaus Dethloff

Copyright © 2016 Katie L. Crandall et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An extended version of the -vector form for the *ω*-equation that includes diabatic (in particular latent) heating in the -vector itself is derived and tested for use in analyzing the life-cycle of a midlatitude cyclone that developed over the central United States during 24–26 December 2009. While the inclusion of diabatic heating in the -vector *ω*-equation is not unique to this work, the inclusion of diabatic heating in the -vector itself is a unique formulation. Here it is shown that the diabatic -vector gives a better representation of the forcing contributing to the life-cycle of the Christmas Storm of 2009 using analyses derived from the 80-km NAM.

#### 1. Introduction

For more than 60 years, quasigeostrophic theory (QG theory) has provided the underlying basis for techniques that explain the existence or evolution of a wide range of atmospheric phenomena including cyclones (e.g., [1–3]) and blocking anticyclones (e.g., [4–8]). QG theory has also been used and continues to be used, as a guiding principle in weather analysis and forecasting (e.g., [1, 9–11]). QG theory represents a scaling of the primitive equations using the concept of approximate equality between the horizontal pressure gradient force and Coriolis (inertial) force in a two-dimensional atmosphere. QG theory also represents an acknowledgement that atmospheric circulations are three-dimensional but that the vertical components of these circulations are much weaker than their horizontal counterparts, and these serve to restore geostrophic and hydrostatic balance. In the earlier applications of QG theory (e.g., [1]), the atmosphere is assumed to be adiabatic, which neglects the role of diabatic processes (such as latent heat) and surface friction. Also, in model simulations or in diagnostic studies, large-scale vertical motions are typically calculated subject to an initially QG balanced environment (e.g., [12]).

Given advances in the understanding of the lifecycles of atmospheric phenomena as well as advances in computing power, there are many studies which have examined the role of forcing neglected in earlier studies using QG equations in midlatitude cyclones (e.g., [13–18]). These have included processes such as boundary layer friction, boundary layer sensible heating, or latent heat release. These have even been included in the study of large-scale phenomena such as blocking anticyclones (e.g., [19–21]), more specifically midlatitude ridging due to lower tropospheric diabatic heating. By including processes or forcing such as diabatic heating or friction in the study of these phenomena, a better understanding of their life-cycle evolution has been gained. This has resulted in better model formulations of, for example, convection (e.g., [22]), as well as providing better guidance to operational forecasters.

While many of the diagnostic tools used in meteorological analysis were developed in full form, such as the omega equation (e.g., [23]) or the Zwack-Okossi vorticity tendency equation (e.g., [5, 16]), they have primarily been used in QG form in many studies (e.g., [24–27]). An increase in computing power has been one factor in inspiring the use of more complete forms of these diagnostic equations (e.g., [4, 13, 16, 20, 28]), and these publications, as well as others, initially refer to the QG-equations that include latent heat and/or friction as “extended” QG forms. In recent years, these “extended” equations are referred to as QG forms, dropping the “extended” notation. The increase in computing power has also led to the development of new expressions, such as the ageostrophic vorticity tendency equation (e.g., [18]). These new diagnostic quantities have allowed the user to examine the role of traditional atmospheric forcing processes using a more complete framework.

The role of diabatic heating in forcing ascent (e.g., latent heating) has long been included by researchers for use in the diagnostic equations cited in examples above. Other examples include the studies of [29], who developed and [30] later used a -vector form of the -equation. Then [30] further included latent heating on the right hand side of the equation as an additional forcing process. In [30], a cyclone case occurring over the Iberian Peninsula was studied, and the latent heating was included in their -vector equation in order to examine the role of this process in the cyclone. Both studies demonstrated that the inclusion of latent heating provided a better estimate for the divergence field that is associated with vertical motion. Thus, in both of their work, the divergence of was calculated separately from the Laplacian of latent heating in order to determine the role of this process in cyclone development.

The standard -vector equation [25] is used heavily in operational analysis and forecasting as it amalgamates important QG forcing processes into one variable (e.g., [17, 25, 26]). In the standard formulation of the -vector, the differential vorticity advection and the Laplacian of the temperature advection are combined to form a term that contains the advection of the temperature by the gradient of the wind field. This has some important advantages, such as eliminating “overlap” between the differential vorticity advection and the Laplacian of the temperature advection and the fact that the forcing is Galilean invariant (e.g., [31]).

Diabatic heating of any kind is not included regularly in any formulation of known to the authors. Since diabatic heating (in particular latent heating) has been shown to be a contributor to the development of phenomena such as midlatitude cyclones, it is often included in diagnostic studies using, for example, the omega equation. Since there is a -vector form of the omega equation, it would be useful to develop a -vector expression that includes the contribution of diabatic heating inside the one variable . Thus, this is the goal of the work here, and the details will be presented in Section 2. In Sections 3 and 4, we perform a case study of a synoptic-scale cyclone where diabatic heating (in particular tropospheric latent heating) was an important process and demonstrate the effectiveness and utility of the diabatic -vector. Section 5 will summarize the work done here and present our conclusions.

#### 2. Data and Methods

##### 2.1. Derivation of an Extended -Vector Form of the -Equation and a Diabatic -Vector

The derivation of an extended -vector is similar to the QG version found in [32]. This derivation begins with a quasigeostrophic form of the Navier-Stokes equations ((1a), (1b), and (1c)) here:where and are the zonal and meridional wind components, respectively. The subscripts “” and “” represent a geostrophic or ageostrophic wind, respectively. The Coriolis parameter is and the change in the Coriolis parameter in the meridional direction is represented by . Friction is represented by “.” The complete derivation of the -vector can be found in [10].

Then, following, for example, [10, 26, 30] and others, we arrive at (2a) and (2b) the final extended -vector form: whereIn (2a), (2b), and (2c), is used for the diabatic heating instead of the traditional notation of in order to avoid confusion with the -vector. Taking the partial derivative of (2a) and (2b) with respect to (), adding the two parts together, and using continuity in the formresult in (4a), (4b), and (4c). Consider the following:where is now comprised of the traditional -vector: The result is a traditional form of the -vector relationship ((4b) and (4c)) derived by [25] originally and found in [32] in QG form, respectively. In (4a), Term A is the divergence of , Term B is the “Beta Term” (meridional difference in the Coriolis parameter), Term C is the diabatic heating term, and Term D is the friction term. Further, for the “extended” forms derived by those cited in section one, they also state that the observed winds can be used in the calculation instead of their geostrophic values.

However, one of the results of the [25] -vector derivation was that the differential vorticity and the Laplacian of the temperature advection terms in the -equation were combined. Since the Laplacian operator is the divergence of the gradient operator, or , (4a), (4b), and (4c) can be rewritten as;where now has the componentsand the diabatic heating term is now clearly part of the -vector formulation ((5b) and (5c)) and the divergence of and not a separate term as in [29, 30]. Thus, the goal of this study will not be to demonstrate primarily the importance of diabatic heating in contributing to cyclone development, since that has been established by many researchers, but the utility of a diabatic -vector in an operational context. Nonetheless, it will be compared to the traditional -vector as a point of reference. It is also noted here that this study only examines the latent heating in the diabatic -vector and sensible and radiational heating/cooling process are neglected.

##### 2.2. Data and Analysis

Numerical output from the North American Mesoscale (NAM) Eta model was used to make the calculations of the diabatic -vector and its components. In particular, a thinned, 80 km grid was employed for the QG diagnostics as suggested by [33]. While this is relatively coarse resolution compared to what is used operationally today, these data were readily available in-house. The operational resolution at the time of this storm was 12 km out to 84 h. Output was taken from the run initialized at 1200 UTC 24 December 2009. At this time, a well-developed cyclone was already present over the southern United States, as suggested by both the surface analysis (showing a mature cyclone) and the satellite signature of a comma cloud in the southern plains (see Figure 1).