Title: Evaluation of a limited-area energy budget cycle of an extratropical storm under Lagrangian and Eulerian frameworks

: To conceptualize the uncertainties regarding the mechanisms of extratropical cyclones (EC), a study of their energy cycle can provide key information of their fundamental structure. This study applies a set of equations built from earlier works for a limited-area energy decomposed into temporal mean and deviations. It compares the results obtained with a reference frame that tracks an EC through its eddy kinetic energy with those obtained with a larger but fixed frame. A specific storm that occurred throughout the period of December 10-18 th 2004 and simulated by the Canadian Regional Climate Model (CRCM – version 5) was studied. Results support the notion that the moving reference results in larger amplitudes for all temporal deviation components of the cycle than for the fixed reference. A time tendency analysis of the energetic reservoirs reveals noteworthy phases in the storm’s energy, with an increase and decrease occurring during the periods of 10-14 December and 14-18 December, respectively. The energy budget is overall fairly well balanced, with the exception of a lateral boundary term, , with considerable negative values; this term exhibits a spatially larger scale than the other contributions in the EC. An evaluation of the sensibility of the tracking scheme related to its size and positioning was also performed to determine its influence on the boundary term

1. Introduction 1 kinetic energy (ZKE) by barotropic processes occurring mainly in the stage of cyclolysis (Kuo 28 1951;Starr and White 1951;van Mieghem 1952). Oort (1964) was the first to quantify Lorenz's 29 cycle for annual-mean conditions over the northern hemisphere. 30 Following Lorenz's seminal work, numerous studies have been carried out to examine the energy 31 of the atmosphere (van Mieghem 1955;Wiin-Nielsen 1967;Peixoto and Oort 1974;Michaelides 32 1987, Yeon andMaeng 2013; to name just a few). By using Lorenz's theory, Oort (1964) and 33 Dutton and Johnson (1967) have shown the importance of baroclinic conversion in the 34 development of storms, which curtailed an earlier hypothesized importance of barotropic 35 conversion (Smith 1969). 36 In today's climate research, atmospheric energy budgets have proven themselves to be pertinent 37 tools (O'Gorman and Schneider 2008) especially when attempting to target EC (Muench 1965;38 Smith 1969;Johnson 1970;Marquet 2003). Most notably, they have the potential to reveal the 39 specific processes that occur during the lifetime of EC (Chang et al. 2002); such studies, however, 40 require the use of limited-area energy budgets, which differ considerably from their global 41 counterparts. 42 In limited-area energy budgets emerge additional boundary flux terms that represent energy 43 transferred between the regional and global systems. New perspectives into the study of regional 44 atmospheric energetics have emerged from the work of Orlanski et al. (1991Orlanski et al. ( ,1993aOrlanski et al. ( , 1993bOrlanski et al. ( , 1995 who underlined the role of ageostrophic geopotential fluxes in the development of baroclinic 46 waves. Such fluxes cancel out in global energy budgets and hence could not have been identified 47 algorithms already exist to identify storm tracks such as band-pass filters and feature tracking, but 80 few have ever considered using EKE and AE through a Lagrangian framework (Wing 2009). 81 Vincent and Chang (1975) were amongst the first to attempt a Lagrangian approach: they used 82 contour lines of pressure to evaluate KE located within it. Michaelides et al. (1999) used what they 83 called a "semi-Lagrangian approach" by considering a volume bounded by the 1000 and 100 hPa 84 pressure levels and moving horizontally with the cyclone track. Recently Papritz and Schemm 85 (2013) followed the movements of an idealized baroclinic wave by defining a squared-perimeter 86 around the 100 Jm -2 contour lines of KE. The authors of these studies concurred that the magnitude 87 and the role of the different energy conversions are most evidenced using a Lagrangian reference 88 system. 89 The paper is organized as follows. Sect. 2 will present the methodology of the procedure, including 90 the data used as well as the energetic equations and the Lagrangian computational procedure.
The use of the lower case and upper case serve to differentiate between instantaneous and 108 climatological values, the former being of most interest in this study. 109 The current study will focus on the transient-eddy energetics and each energy reservoir and 110 conversion terms will have the subscript 'TV' for time variability as depicted by Fig. 1.  111 For the study of an individual storm, the evolution of two energy reservoirs is described by the 112 following equations: 113 12  TV  TV   TV  TV  TV corresponds to the available enthalpy due to the time variations of temperature, calculated from 118 the instantaneous deviation from the monthly mean. It is divided by r T that corresponds to the 119 reference state (Marquet 1990); for the current study, the reference temperature has been 120 approximated to 262 K.
The terms on the RHS of (1.4) are evaluated as: 135 are computed by using a centred finite-difference method 174 over 6-hourly intervals. In this work the shape and area of the diagnostic domain over which the 175 local time derivatives are computed are kept constant. 176 During the lifetime of a specific storm, the time tendencies initially grow and then decay; hence 177 integrated over sufficient time intervals and spatial domain, they should vanish. In practice, 178 however, the presence of computational errors, physical approximations, interpolations and the 179 inevitable finite spatial and temporal computational limits prevent obtaining exactly vanishing 180 values, as well as an exact correspondence between the left-hand side tendencies L and the sum of 181 the right-hand side contributions (R). A comparison between L and R remains nonetheless a useful 182 method to evaluate the accuracy of the computed energetics. Spatial derivatives are approximated 183 as centred finite differences and vertical integrals are computed using the trapezoidal rule after 184 eliminating contributions below the surface, as in CNL16. 185 A Lagrangian diagnostic domain was defined as a rectangle within the CRCM5 computational 186 grid, tracking the selected storm's path. A rectangle was employed to facilitate programming and 187 to preserve a nearly constant surface area during the tracking process, which also eases the 188 interpretation of boundary fluxes. The tracking process was divided into several steps that may be 189 summarized as follows. The tracking rectangle shape and size was selected such that vertically 190 integrated TV a and TV k fields would remain essentially confined within the tracking rectangle. It 191 was noted that simply using TV k was sufficient to shape the reference as it encompassed both Scheme (CLASS) v3.5 (Verseghy 2000(Verseghy , 2009 and an interactive column lake module is also 208 included . The CRCM5 simulation is configured for a 0.44 o rotated latitude 209 and longitude grid mesh, with a free domain of 260 by 160 grid points in the horizontal, covering 210 Canada, USA, Greenland, the north of Mexico, and neighbouring oceans (Fig. 2). The simulation 211 uses 56 terrain-following hybrid levels in the vertical, up to 10 hPa, and the timestep is 12 minutes. 212 Output data is archived at three hourly intervals, interpolated on 19 pressure levels, but energy 213 diagnostics will only be evaluated from 1000 to 150 hPa. 214 The CRCM5-simulated fields rather than the reanalyses fields will be used for the energetic 215 calculations for 2 main reasons: the simulation provides superior time and spatial resolutions, and 216 several fields required by the energy budget are not routinely available in the reanalyses. 217

Storm selection and synoptic overview 218
In order to facilitate testing the proposed procedure for carrying energy budget calculations in a 219 Lagrangian framework, it was deemed preferable to select a rather mainstream extratropical storm 220 that did not merge or split during its lifetime. A rectangular area was selected that would allow 221 resolving the storm at the various stages of its life cycle within the confinement of the 222 computational domain, which would minimize artefact due to limits imposed by it, as would occur, 223 for instance, if part of the storm were to leave the domain while being tracked. The selected storm 224 has been documented as a significant winter storm due to its heavy precipitation, freezing rain and 225 flooding across the northeastern USA and eastern Canada (NWS 2010). Weather charts for daily 226 values of sea-level pressure, 850 hPa temperature and 500 hPa geopotential height are shown in 227 The interest of employing a Lagrangian reference framework for computing energetics becomes 265 apparent in Fig. 4 and 5 that compare the energetics using Lagrangian and Eulerian frameworks. 266 This is consistent with previously cited atmospheric energetics studies using a similar 272 methodology (Vincent and Chang 1975;Michealides et al. 1999;Papritz and Schemm 2013). where there are significant temperature anomalies (panel in Fig. 6a). These anomalies can be more 290 easily understood when comparing the panels Fig 6 with the 850 hPa temperature maps of the 291 panels in Fig 2. There are noticeable alignments between the areas of intense a TV around the low-292 pressure center (panel Fig. 6a) with the areas consisting of cold and warm air advection. Near 14, 293 00 UTC, two peaks of a TV are noticeable, associated with the warm and cold sectors of the storm.

294
Later the warm air advection dominates as the storm occludes and moves northwestward. The 295 spatial pattern of TV k shown in Fig. 6b closely reflects the jet stream (not shown) and it is initially 296 strongest to the west of the trough. The initial peak of TV k is located west of the developing low-297 pressure centre, and it later shifts towards its centre by 13, 00 UTC. Up until the dissipation phase, 298 the energy east of the trough is included within the storm's energy, but as the trough expands near 299 the end of the storm's life, the energy located over it no longer becomes relevant. results are similar to that of CNL16 and Nikiéma et al. (2017). For example, the approximate 307 duality in magnitude and pattern between TV c and A c is noteworthy in Fig. 7a. As seen in Fig. 8a,  308 there is also an important contribution of TV g that peaks in the mid-troposphere, where clouds and 309 precipitation form in extratropical cyclones, reflecting the release of latent heat energy. This would 310 hence corroborate the findings of previous work that noted that latent heat release is a strong 311 component of transient-eddy energy generation for mid-latitude cyclones (Danard 1966;Bullock 312 and Johnson 1971;Michaelides 1987). A negative product between Q and T  can lead to negative 313 values of TV g which occur near the surface (Fig. 8a). This is due to boundary-layer heat flux that 314 is frequently occurs during winter (Nikiéma et al. 2017). During the second period, the contribution 315 of TV g slightly extends downwards to 800 hPa and has greater negative magnitude near the surface. 316 There are also notable differences when comparing the baroclinic conversion terms. In both 317 periods, there are asymmetrical distributions of conversion and could consequently represent the ageostrophic convergence of energy, which 334 might act as an initial amplifier for the disturbance, as noted by Orlanski and Sheldon (1995). Their 335 study showed ageostrophic geopotential flux convergence of downstream development that occurs 336 in the upper troposphere and is intrinsically linked with energy transfers between consecutive 337 cyclones, the study of which however is beyond the scope of this work. Most importantly, one 338 must note that these early positive values of occurrences, the horizontal transfer is stronger than the vertical one, supporting the results of van 375 Mieghem (1955). This explains why this conversion term is known as barotropic, because even in 376 the absence of baroclinicity, K c remains strong. 377 The term TV k h is shown (Fig. 8d)  contribute to the overall negative tendency in the last three days of the storm (Fig. 9d) Lagrangian diagnostic domain at the northeastern boundary (Fig. 9d). 435

Summary and conclusion 436
The purpose of this work was to compare regional atmospheric energy budget for an EC occurring 437 over North America under two different perspectives: a Eulerian framework where budget 438 diagnostics are computed over a wide and fixed domain, and a Lagrangian framework where these 439 computations are performed over a smaller but mobile domain that tracks the system through its 440 eddy kinetic energy. A hypothesis was put forward that the latter method is the optimal approach 441 to adequately interpret an EC because the analysis would most notably reveal stronger energetics 442 as well as otherwise concealed properties as would be the case under a Eulerian framework.

478
Ultimately, the Lagrangian framework fulfils its purpose by providing a realistic and accurate 479 depiction of the extratropical cyclone. Through an energy budget, the multiple aspects of a storm 480 are reduced to a common denominator and their aspects are more effectively understood. Many 481 studies have limited themselves with using a fixed and large domain to evaluate its energetic 482 behaviour. Although useful enough to understand the underlying principles of how a storm evolves, 483 this type of reference lacks the precision that a Lagrangian framework can provide, as was 484 illustrated in this study. As this is a seldom-used method in the study of atmospheric energetics, 485 its potential and the amount of information that can be exploited from it are considerable. Even 486 though this study focused on a synoptic-scale system, this kind of framework can prove itself to 487 be particularly useful for tropical cyclones or mesoscale convective systems. Further use of this 488 method could be found as an automatic storm-tracking algorithm. With the increasing interest in 489 machine learning, a supervised machine-learning algorithm could be devised to distinguish and 490 connect the areas with high TV a and TV k over multiple time steps and devise a tracking system 491 from these formed connections. Another useful way of exploiting this method is through the 492 tracking of a large ensemble of storms and computing various statistical parameters from their 493 compiled energetic values. This type of study could then be extended into a comparison between 494 the energy cycle of storms in the current climate versus that of a changed climate under different 495 emission scenarios. However, and like all storm-tracking methods, this method has limitations in 496 the sense that certain parameters are subjectively based. The prime example of this would be the 497 imposed threshold that distinguishes areas of high energetic values to the low ones. It would be 498 possible for certain systems to exist without ever going pass this threshold value, making them 499 unnoticeable to the algorithm. This problematic would be but one among many others that would 500 need to be considered if such a scientific endeavour were to ever occur. 501 502 503 504 505 List of figure legends Fig. 1 Limited-area energy cycle for transient eddies, following the methodology developed by NL13 and applied for temporal disturbances by CNL16. Fig. 2 Instantaneous maps of 850-hPa temperature (in colour), with 500-hPa geopotential height (in red contours) and mean sea level pressure (in black contours), from 09 to 16 December 2010, all at 00 UTC. Units: temperature ( o C), sea level pressure (hPa), geopotential height (dam). Images are rotated -90 degrees.      (full lines) and Eulerian (dotted lines) reference frames, averaged between December 10-14 th (the growing period; left-hand side) and December 14-18th (the decaying period; right-hand side).