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    Conceptual diagram of a barotropic growth signature sensitivity with respect to initial condition zonal wind at jet level. Gray lines are isotachs, with a “J” indicating the jet core. Vectors indicate relatively stronger zonal flow within the jet core. Sensitivity with respect to zonal flow is indicated with shaded regions representing areas of positive (red) and negative (blue) sensitivity.

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    Sensitivity with respect to 0.8750-sigma zonal flow (shaded every 0.1 Pa s m−1, cool colors negative) and 0.8750-sigma zonal flow (contoured every 2 m s−1, negative contours dashed) at model initialization for (a) a representative PBS case (Cosme), (b) a representative SBS case (Bud), and (c) a representative NBS case (Linda). All fields are valid at model initialization.

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    Difference test for efficacy of inverse Laplacian assuming zero-boundary conditions. (a) Sensitivity with respect to zonal flow at 0.8750 sigma for Tropical Storm Alma (2008), shaded every 0.1 Pa s m−1, cool colors negative. (b) Sum of sensitivity with respect to nondivergent and irrotational components of zonal flow. (c) Shading as per (a), with difference between (a) and (b) contoured in black every 0.02 Pa s m−1, negative contours dashed.

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    Storm-centered composites of sensitivity with respect to (left) zonal and (right) meridional flow at the initial time (shaded every 0.5 Pa s m−1, cool colors negative) and zonal flow (contoured every 1 m s−1, negative contours dashed) for (a),(b) PBS cases, (c),(d) SBS cases, and (e),(f) NBS cases.

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    Storm-centered composite of 0.8750-sigma zonal flow (positive values contoured in black every 1 m s−1), meridional gradient of zonal flow (negative values contoured in magenta every 1 × 10−5 s−1), and meridional gradient of absolute vorticity () (shaded every 1.0 × 10−10 m−1 s−1, cool colors negative) for (a) PBS cases, (b) SBS cases, and (c) NBS cases at the initial time. Composite south–north plot of (d) zonally averaged meridional gradient of zonal flow and (e) zonally averaged meridional gradient of absolute vorticity () for PBS cases (blue), SBS cases (green), and NBS cases (red). The “X” along the abscissa marks the location of the composite storm center, and the “J” represents the mean latitudinal position of the zonal jet maximum at this level.

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    Sensitivity with respect to initial condition temperature (shaded every 0.1 Pa K−1, cool colors negative) and sensitivity with respect to initial condition vertical motion (contoured every 2 × 10−2 Pa s m−1, negative contours dashed) for (a) the 0.7750-sigma level and (b) a west–east cross section from 0.9750 to 0.3250 sigma. (c) Profile of correlation coefficient between sensitivity with respect to initial condition temperature and sensitivity with respect to initial condition vertical motion on each sigma level.

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    Vertical structure of mean optimal perturbations for each category. Mean profiles of (a) percent of total energy added to each sigma level and (b) amount of energy added to each sigma level (PBS = blue, SBS = green, NBS = red). The gray line corresponds to a profile of zonal-average flow through the core of the composite jet amongst all cases. Magenta line along y axis highlights levels where differences between PBS and NBS profiles surpass a Student’s t test at 95% confidence. West–east cross section of perturbation temperature (shaded every 0.1 K, cool colors negative) and perturbation vorticity (contoured every 1 × 10−5 s−1, negative contours dashed) for (c) PBS mean, (d) SBS mean, and (e) NBS mean.

  • View in gallery

    (a) Composite profiles of summed perturbation kinetic (solid lines) and available potential (dashed lines) energy on each sigma level for the PBS composite (blue), SBS composite (green), and NBS composite (red). (b) Storm-centered composite of 0.8750-sigma optimal temperature perturbations for PBS cases (shaded every 0.05 K, cool colors negative), composite vorticity (magenta contours every 1.0 × 10−5 s−1, positive values only), and composite zonal flow (black contours every 1 m s−1, positive values only). (c) Cross section SW–NE through Tropical Storm Alma (2008) forecast at 9 h into the simulation. Relative vorticity (shaded every 4 × 10−5 s−1, positive values only), perturbation zonal flow from the (u, υ)-perturbation experiment (black contours every 3 m s−1, negative contours dashed), and perturbation zonal flow from the (T, q)-perturbation experiment (magenta contours every 3 m s−1, negative contours dashed). (d) As in (c), except perturbation meridional flow is contoured.

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    Storm-centered composites of localized energy generation rate of wind perturbations from (left) basic-state shearing deformation and (right) basic-state stretching deformation at the 0.8750-sigma level. Composites are shown for (a),(b) the PBS category, (c),(d) the SBS category, and (e),(f) the NBS category. Energy generation rate is shaded every 4 J Kg−1 s−1 (cool colors negative). Basic-state vorticity is contoured in magenta every 1 × 10−5 s−1 (positive values only), and basic-state zonal flow is contoured in black every 1 m s−1 (positive values only).

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    Composite profiles of summed energy growth rate normalized by summed initial perturbation energy by sigma level for (blue) PBS cases, (green) SBS cases, and (red) NBS cases.

  • View in gallery

    Storm-centered composites of sea level pressure (contoured every 1 hPa) and difference in sea level pressure between perturbed and unperturbed simulation (shaded every 0.5 hPa, cool colors negative) at (left) 24 h and (right) 48 h for (a),(b) PBS cases, (c),(d) SBS cases, and (e),(f) NBS cases.

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    Response function box-centered composite across all cases of change in perturbation pressure at 24 h into the simulation for (a) (u, υ, T, q)-perturbation test, (b) (u, υ)-perturbation test, and (c) (T, q)-perturbation test. Difference in perturbation pressure between perturbed and unperturbed simulations is shaded every 20 Pa (cool colors negative). The magenta box outlines the confines of the response function box. (d) Average observed change in response function across all cases for (left)–(right) (u, υ)-perturbation test, (T, q)-perturbation test, the sum of these functions, the (u, υ, T, q)-perturbation test, and the desired change in the response function defining optimal perturbations. Values are given in hPa.

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Identifying a Barotropic Growth Mechanism in East Pacific Tropical Cyclogenesis Using Adjoint-Derived Sensitivity Gradients

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  • 1 Space Science and Engineering Center, University of Wisconsin–Madison, Madison, Wisconsin
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Abstract

The eastern Pacific tropical cyclone basin is typified by a low-level westerly jet with the main development region residing on its northern, cyclonic-shear side. The persistent meridional shear of the zonal flow associated with the jet allows for the possibility of barotropic conversion of energy from the mean state into the kinetic energy of vortices—possibly contributing to tropical cyclogenesis, but this is difficult to quantify by perturbing the model based on intuition since there is no guarantee that perturbations will favorably interact with the jet to facilitate cyclogenesis.

Here, sensitivity gradients of vortex intensity through cyclogenesis are calculated for a set of cases spanning from 2004 to 2010 and are interpreted dynamically to determine which cases have sensitivities describing structures that can grow barotropically from the low-level jet. The adjoint model is run with adiabatic physics linearized about a basic state that contains moist convection. Optimal perturbations derived from these sensitivities are inserted into the model to observe the impact. Roughly 34% of observed cases exhibited structures in sensitivity to zonal flow that strongly imply barotropic growth, while about 21% exhibited no such structures. The remainder (roughly 45%) exhibit some reliance on barotropic growth. Cases with sensitivities exhibiting strong barotropic growth structures are typified by low-level westerly jets with larger meridional shear. In these cases, optimal perturbations require less initial energy to increase vortex intensity by a specified amount, the energy is more strongly focused at jet level, and the localized energy growth rate of perturbations is most efficient.

Corresponding author address: Brett T. Hoover, Space Science and Engineering Center, University of Wisconsin–Madison, 1225 W. Dayton St., Madison, WI 53706. E-mail: brett.hoover@ssec.wisc.edu

Abstract

The eastern Pacific tropical cyclone basin is typified by a low-level westerly jet with the main development region residing on its northern, cyclonic-shear side. The persistent meridional shear of the zonal flow associated with the jet allows for the possibility of barotropic conversion of energy from the mean state into the kinetic energy of vortices—possibly contributing to tropical cyclogenesis, but this is difficult to quantify by perturbing the model based on intuition since there is no guarantee that perturbations will favorably interact with the jet to facilitate cyclogenesis.

Here, sensitivity gradients of vortex intensity through cyclogenesis are calculated for a set of cases spanning from 2004 to 2010 and are interpreted dynamically to determine which cases have sensitivities describing structures that can grow barotropically from the low-level jet. The adjoint model is run with adiabatic physics linearized about a basic state that contains moist convection. Optimal perturbations derived from these sensitivities are inserted into the model to observe the impact. Roughly 34% of observed cases exhibited structures in sensitivity to zonal flow that strongly imply barotropic growth, while about 21% exhibited no such structures. The remainder (roughly 45%) exhibit some reliance on barotropic growth. Cases with sensitivities exhibiting strong barotropic growth structures are typified by low-level westerly jets with larger meridional shear. In these cases, optimal perturbations require less initial energy to increase vortex intensity by a specified amount, the energy is more strongly focused at jet level, and the localized energy growth rate of perturbations is most efficient.

Corresponding author address: Brett T. Hoover, Space Science and Engineering Center, University of Wisconsin–Madison, 1225 W. Dayton St., Madison, WI 53706. E-mail: brett.hoover@ssec.wisc.edu

1. Introduction

Sensible weather in the eastern Pacific is influenced by the existence of a lower-tropospheric (surface to ~850 hPa) westerly jet centered around 5°N (Poveda and Mesa 2000); this jet feature in the deep tropics of the eastern Pacific is manifest as a zonally elongated region of enhanced westerly flow with a significant meridional gradient of the zonal wind on either side. The jet presents the possibility for the growth of tropical disturbances into tropical cyclones (TCs) through barotropic energy conversion of shear to eddy kinetic energy on the cyclonic-shear side (Guinn and Schubert 1993; Maloney and Hartmann 2001; Hartmann and Maloney 2001), though this mechanism has been shown to not be a necessary condition for eastern Pacific tropical cyclogenesis (Davis et al. 2008). The conversion of energy in the shear of the mean state to eddy kinetic energy has been postulated as a potential driving mechanism in east Pacific tropical cyclogenesis (Lipps 1970), especially during intertropical convergence zone (ITCZ) “breakdown” scenarios where the ITCZ transitions from a zonally elongated convective feature to one or more discrete tropical vortices (Ferreira and Schubert 1997).

Significant questions remain unanswered regarding the role of barotropic energy conversion in the genesis of TCs in this region, such as the overall importance of this process compared to other mechanisms (air–sea interaction, baroclinic growth, etc.) for individual TCs. Furthermore, while success in TC genesis and intensity prediction in numerical weather prediction (NWP) models has lagged significantly behind other aspects of TC prediction such as track (DeMaria and Gross 2003), the genesis of some TCs has been predictable with high accuracy several days in advance; it is possible that the predictability of these TCs is due to features of their environment that are well resolved by NWP models (Davis et al. 2008).

Given a function of model-state output (a “response function”), one can employ the adjoint of a linearized NWP model to calculate the gradient of the response function with respect to the model state. This gradient is the sensitivity of that response function to small perturbations to the model state at earlier times (Errico 1997) or to derived variables of the model state (Kleist and Morgan 2005). These sensitivity gradients provide insight into the dynamics of a modeled atmosphere that are otherwise difficult or impractical to obtain. Often an attempt is made to determine the sensitivity of a model forecast to a particular feature of the initial state by prescribing perturbations to that feature based on intuition and then observing the result on the forecast (e.g., Komaromi et al. 2011); this is sometimes characterized as a more robust “sensitivity study” by including several different kinds of perturbation experiments in an attempt to make the results seem less anecdotal. The advantage of an adjoint model is that the sensitivity gradients provide the necessary a priori information about the impact any perturbation of a particular variable in a particular geographic region would have on the chosen response function. The costly and ultimately still arbitrary technique of perturbing the initial conditions in several different ways is avoided, and instead the sensitivity gradients can be used to define perturbations that will have an impact on a specific aspect of the forecast (Errico 1997; Blessing et al. 2008).

Adjoint models have been used in the past to investigate sensitivity of sea level pressure (Langland et al. 1995) or low-level vorticity (Vukićević and Raeder 1995) in midlatitude cyclones to infer the importance of various synoptic-scale features of the cyclone’s environment to the intensity of that cyclone. Application of adjoint models to TC development has largely been restricted to sensitivity of TC steering (Peng and Reynolds 2006; Wu et al. 2007, 2009; Chen et al. 2009; Hoover and Morgan 2010); this is at least partially due to the constraints of linearity and simplified moisture physics that limit the applicability of adjoint models to tropical dynamics, though adjoint techniques have been applied to problems surrounding TC structure and genesis in more recent years as models have grown more sophisticated (Doyle et al. 2012; Lang et al. 2012).

In this study, the adjoint of a NWP model is employed to investigate the importance of barotropic growth to east Pacific TC genesis along the low-level westerly jet. The goal is to identify a “signature” in the adjoint-derived sensitivities that indicates the importance of barotropic growth of vorticity at the expense of the background shear. As this analysis is primarily concerned with the potential impact of small perturbations with respect to the synoptic-scale, dry dynamics of this possible genesis mechanism, an adjoint model is uniquely suited to this task. A description of the model used is provided in section 2. Section 3 provides a description of the methodology used for defining the TC intensity response function, describing a “barotropic growth signature” in the sensitivity gradients, and defining optimal perturbations to the model initial state. Results of the analysis are described in section 4, and conclusions and directions for future research are given in section 5.

2. Model

The Pennsylvania State University (PSU)–NCAR Mesoscale Model (MM5) version 2 and its adjoint (Zou et al. 1997) are used. The MM5 model is a nonhydrostatic, limited area, primitive equation model that uses as its vertical coordinate a terrain-following sigma coordinate. For all sensitivity calculations performed, the nonlinear version of MM5 is used to create a basic state about which the tangent linear model (TLM) and adjoint model are linearized. For the TLM and adjoint integrations, the basic state is updated every time step. The model is run with the Grell cumulus convection scheme. Model simulations are run at 18-km grid spacing, use a 36-s time step, and are initialized with NCEP 1° × 1° final reanalysis data (ds083.2)1 for cases before November 2006, while model simulations run after that time are instead initialized with higher-resolution GFS 0.5° × 0.5° analysis fields from the Historical Unidata Internet Data Distribution Gridded Model Data archive (ds335.0).2

The MM5 adjoint model is run with a bulk PBL scheme. The effect of radiative cooling on the long-term mean tendency on temperature is switched off, as per the suggested settings for a sensitivity calculation (Zou et al. 1997). While there exist some capabilities within the MM5v2 adjoint model to incorporate the physical effects of moisture, these physics options have a tendency to cause sensitivity gradients to grow exponentially or otherwise return unrealistic results. These physics have been switched off in the adjoint and, instead, the adjoint model approximates the nonlinear MM5 with the physics of dry convection only. The impact of moist physics on the adjoint-derived sensitivity gradients is not entirely missing, however, since the sensitivity gradients are integrated along a basic-state nonlinear trajectory that includes the full moist physics (Kleist and Morgan 2005). Since the focus of this study is on a process for vortex growth that does not depend upon moisture physics, this simplification was deemed both acceptable and necessary. An examination of the inclusion of moist-physical processes in an adjoint model that includes the effects of moist convection is an area of further study.

3. Methodology

a. Computing sensitivity gradients

Simulations of a set of 53 east Pacific tropical cyclogenesis cases are performed for 48-h trajectories encompassing the 24 h leading up to declaration as a tropical storm and 24 h after declaration. For each case, sensitivity gradients are computed for a response function defined as the perturbation pressure at the bottom (sigma) level in a roughly 250 km × 250 km box (15 grid points on a side) centered on the (24 h) forecast position of the vortex:
e1
Here, p′ is the perturbation pressure and D represents the region containing all of the grid points in the box, indexed zonally by i and meridionally by j. The negative sign is included because an intensity increase is associated with a pressure decrease and vice versa; thus, the response function describes vortex intensity rather than simply pressure. The pressure-based response function, representing the change in mass of the atmospheric column above the TC, is similar in form to previous work examining the sensitivity of midlatitude cyclogenesis (Langland et al. 1995) and, unlike other options (e.g., low-level vorticity or kinetic energy of the wind field), the pressure is a smooth, (roughly) axisymmetric field with a defined minima, providing a simple and direct relationship between R and TC intensity. The adjoint model is initialized at the 24-h model state and is integrated backward to produce sensitivity gradients with respect to the model initial conditions. Simulations using the nonlinear NWP model are allowed to evolve for the following 24 h for the purpose of evaluating the impact of initial condition perturbations on the TC forecast beyond time of declaration (see section 3c, below). Table 1 is a list of the cases used, with the initialization time and initial condition dataset used.
Table 1.

List of storm names, model initialization times, and model initial condition datasets for all 53 cases 2004–10.

Table 1.

It is important to recognize that the adjoint technique being applied has known limitations, specifically in its ability to account for both nonlinear processes and the effects of moist convection as described in section 2. The sensitivity gradients will therefore not be perfect representations of how perturbations will evolve in the model. Steps are taken to account for the lack of moist physics in the adjoint model when using sensitivity gradients to define perturbations (section 3c), and section 4c details some tests that are employed to determine what sources of error exist owing to the limitations of the adjoint model and an estimate of the size and contribution of those errors. It is noted that although limitations exist that affect the quantitative value of the sensitivity gradients, the gradients also provide significant qualitative value in their structure and orientation relative to the model basic-state fields.

b. Defining a barotropic growth signature and grouping cases

A barotropic growth signature is (subjectively) identified in the sensitivity with respect to zonal flow as a structure that tilts meridionally upshear with respect to the core of the low-level jet, alternating in sign (Fig. 1; see Langland et al. 1995; Farrell and Moore 1992). Perturbations with this tilted structure have the capacity to extract energy from the shear of their environment and experience transient growth. The structure of sensitivities with respect to jet-level zonal flow is analyzed for each of the 53 cases in order to bin the cases into one of three categories. A pattern in sensitivity to zonal flow that predominantly displays the upshear-tilted, alternating-sign pattern of the barotropic growth signature is considered part of the primary barotropic signature (PBS) bin. If no such barotropic growth signature can be identified, the case is designated part of the nonbarotropic signature (NBS) bin. Cases in between these extremes, where a barotropic growth signature may be present but is not the primary pattern in the sensitivity field, belong in the secondary barotropic signature (SBS) bin. An example of a representative case for each bin is provided (Fig. 2). This binning procedure is subjective within the constraints of focusing on sensitivity of zonal flow at jet level; this allows for flexibility, for example, in constructing a binning category for cases in between the PBS and NBS extremes. Composites of sensitivity gradients, basic-state fields, and perturbations are performed storm centered, with each of these three bins composited separately in addition to a composite comprised of all cases. Composites are produced over an 1800 km × 1800 km area (101 × 101 grid points). Of the 53 cases examined, 18 (~34%) were classified as PBS cases, 24 (~45%) as SBS cases, and 11 (~21%) as NBS cases.

Fig. 1.
Fig. 1.

Conceptual diagram of a barotropic growth signature sensitivity with respect to initial condition zonal wind at jet level. Gray lines are isotachs, with a “J” indicating the jet core. Vectors indicate relatively stronger zonal flow within the jet core. Sensitivity with respect to zonal flow is indicated with shaded regions representing areas of positive (red) and negative (blue) sensitivity.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

Fig. 2.
Fig. 2.

Sensitivity with respect to 0.8750-sigma zonal flow (shaded every 0.1 Pa s m−1, cool colors negative) and 0.8750-sigma zonal flow (contoured every 2 m s−1, negative contours dashed) at model initialization for (a) a representative PBS case (Cosme), (b) a representative SBS case (Bud), and (c) a representative NBS case (Linda). All fields are valid at model initialization.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

c. Optimal perturbation analysis

Given sensitivity gradients, one can construct perturbations designed to change the response function by a specified amount, optimized subject to a chosen constraint (Errico 1997; Blessing et al. 2008). We wish to construct optimal perturbations to the initial conditions to intensify each tropical storm vortex at 24 h into the forecast, subject to the constraint that these perturbations minimize an approximate energy norm (Zou et al. 1997) representing the sum of the kinetic and available potential energy norms:
e2
where the first term represents the kinetic energy added to the system with representing nondivergent perturbations to the zonal and meridional flow, respectively, and the second represents the available potential energy. Here, is the Brunt–Väisälä frequency (calculated between every 2-sigma-level slab and averaged between slabs to approximate values on sigma levels; top and bottom slabs are directly assigned to top and bottom sigma levels) and is the square of the temperature in the basic state. It is important to note that the choice of norm when computing optimal perturbations imposes itself on the preferential spatial scale of the resulting optimal perturbations. Singular vectors (describing the fastest-growing perturbations along a model trajectory, relative to a chosen norm) tend to exhibit planetary-scale features when an enstrophy-based norm is used, while an energy- or streamfunction-based norm is small in spatial scale (Palmer et al. 1998). Various norms can produce optimal perturbations that evolve differently and have different growth rates (Snyder and Joly 1998). Here, the definition of an “optimal” perturbation is a perturbation that achieves a desired change in forecast storm intensity (defined by our response function) while utilizing the minimum initial perturbation energy. The goal of these perturbations is to examine the behavior of the modeled TC when perturbations are defined that project onto the observed sensitivity gradients and do so in a way that allows for comparison across different cases that exhibit different magnitudes of sensitivity. Defining an energy-based norm as the constraint for optimization allows for an objective way to modulate the size of perturbations for each case: each case is supplied a perturbation field that is designed to evoke the same change in TC intensity while utilizing the minimum amount of energy for that situation. Thus, the energy norm is appropriate for this purpose. The energy norm is also attractive as an option because of its use in various studies investigating the dynamics of tropical cyclones with energy-norm-based singular vectors (e.g., Peng and Reynolds 2006; Chen et al. 2009; Kim and Jung 2009; Reynolds et al. 2009; Kim et al. 2011) and recognition that the energy norm is a straightforward function of model-state variables, which allows for employment of sensitivity gradients with respect to model-state variables with relative ease (see below).

This energy norm constraint is similar to that used in Zou et al. (1997) for calculating singular vectors with a few key differences. First, there is no “elastic energy” term (pressure is not perturbed). In addition, two changes have been made to the kinetic energy term. Vertical motion is not perturbed, and perturbations to horizontal components of the flow are nondivergent. This is due to the influence of unrealistic sensitivities of vertical motion and consequently the divergent component of the sensitivities to horizontal flow. This is primarily caused by the adiabatic physics of the adjoint model imposing negative sensitivity to vertical motion where positive sensitivity to temperature exists, suppressing convection over the nascent TC vortex [see section 4a(ii) for details].

Given this constraint (2), we can seek to impose perturbations that will change the response function by while minimizing the selected energy norm. Let represent a vector of the initial state that includes . The energy norm can be written as , where is a diagonal matrix of scaling factors reducing perturbation terms and to kinetic energy and T′ to available potential energy [coefficients of Eq. (2)]. Sensitivity gradients provide information on how perturbations will impact the response function: . This can be solved using a Lagrange multiplier method with the Lagrangian (L) given by
e3
Differentiation of L with respect to the Lagrange multiplier yields the constraint
e4
or
e5
Optimal perturbations are yielded by differentiation of L with respect to :
e6
and, for ,
e7

The adjoint model is initialized with a vector of sensitivity gradients with respect to model-state variables at 24 h into the forecast that is all zeros except for sensitivity with respect to perturbation pressure on the bottom sigma level within the response function domain D, where it is set to −1. The adjoint model then computes the sensitivity of R with respect to model-state variables at the initial time. Sensitivity with respect to u, υ, and T is necessary to compute optimal perturbations [Eq. (7)], but some manipulation of the sensitivity to the wind field is required first. It is desirable here to restrict wind perturbations to the nondivergent component only. This is due to some unrealistic behavior in the adjoint-derived sensitivity gradients with respect to the irrotational component of the wind, arising from unrealistic sensitivity with respect to vertical flow and the dynamical link between these two components through mass continuity. This is explained in detail in section 4b.

Calculation of these perturbations requires knowledge of the sensitivity to each variable, which means we must have information about sensitivity to nondivergent zonal and meridional flow, , . This is obtainable by computing the sensitivity with respect to vorticity and then using this quantity to rederive the sensitivity with respect to the horizontal wind field. One can take advantage of the fact that the wind information can be cleanly divided into a nondivergent component described by the vorticity and an irrotational component described by the divergence, then rederive only the nondivergent portion by inverting only the vorticity to recover the nondivergent wind. An analogous calculation is performed on the sensitivity of R with respect to zonal and meridional flow to produce sensitivity with respect to the nondivergent components of the flow only. This is aided by the Gauss–Seidel, successive overrelaxation scheme (Q) used to invert vorticity to recover the components of the flow from the vorticity being self-adjoint:
e8
e9
Let Z be the definition of vorticity as a function of the nondivergent components of u and υ:
e10
Here, the nondivergent components are used because it is understood that the irrotational component of the flow carries no vorticity. The adjoint of this function produces sensitivities with respect to und and υnd as a function of the sensitivity with respect to vorticity:
e11
Take note of the sign reversal in the spatial derivatives. The sensitivity to the nondivergent components of the wind field is described as spatial gradients of the sensitivity with respect to vorticity; this is the adjoint relationship analogous to vorticity being described as spatial derivatives with respect to the nondivergent components of the wind field, as defined in Eq. (10). A similar argument can be made to derive the sensitivity with respect to the irrotational component of the wind field as spatial gradients of the sensitivity with respect to divergence.
Thus, in order to produce sensitivity with respect to the nondivergent component of the wind field, we must first produce sensitivity with respect to vorticity. This can be accomplished via some manipulation of the relationships derived from Eq. (11). Perform the following manipulation:
e12
which yields
e13
and, therefore,
e14
Finally, recognizing that the sensitivity with respect to the irrotational component of the wind field carries no information about the sensitivity with respect to vorticity, we can substitute (u, υ) for (und, υnd) and arrive at an equation for the sensitivity with respect to vorticity derived from the sensitivity with respect to model-state variables:
e15
This is identical to the formulation found in Kleist and Morgan (2005). A similar argument can be made to define the sensitivity with respect to divergence as an inverse-Laplacian function of the sensitivity with respect to the components of the wind field. Equation (15) functions as our operator Q, and as mentioned above, the inverse Laplacian is solved using a Gauss–Seidel successive overrelaxation scheme that is self-adjoint.

The analysis above makes it clear that under ideal circumstances the sensitivity with respect to the nondivergent and irrotational components of the wind field cleanly separate: , where represents the sensitivity with respect to the irrotational component of the wind field. This is true only up to a tolerance permitted by the boundary conditions used to derive these gradients when performing the inverse Laplacian. Sensitivity gradients with respect to vorticity and divergence are calculated assuming zero-boundary conditions, while the response function is defined as close as possible to the center of the grid so as not to allow for large sensitivity near the edges of the domain where this boundary condition could pose problems. An analysis of the difference between and shows that differences are typically an order of magnitude smaller than the magnitude of and are localized to either slightly blunting the magnitude of maxima and minima or distortion of very-low-magnitude sensitivity near the domain boundaries (Fig. 3).

Fig. 3.
Fig. 3.

Difference test for efficacy of inverse Laplacian assuming zero-boundary conditions. (a) Sensitivity with respect to zonal flow at 0.8750 sigma for Tropical Storm Alma (2008), shaded every 0.1 Pa s m−1, cool colors negative. (b) Sum of sensitivity with respect to nondivergent and irrotational components of zonal flow. (c) Shading as per (a), with difference between (a) and (b) contoured in black every 0.02 Pa s m−1, negative contours dashed.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

Wind and temperature perturbations were calculated for , which is roughly analogous to a 1.5-hPa depression of sea level pressure at each point within the response function domain. This modest change in intensity is intended to keep perturbations small, such that they evolve as close to linearly within the model as possible. The goal of calculating these perturbations is to determine if the structure of perturbations depends on the characteristics of the low-level westerly jet and if there are any significant differences between perturbations calculated for PBS cases versus NBS cases. Initial perturbations are allowed to evolve in the nonlinear NWP model for the entire 48-h trajectory. The model state at 24 h allows for evaluation of the efficacy of the adjoint model for defining perturbations that can reliably intensify the tropical storm (Figs. 11 and 12), while the model state at 48 h allows for evaluation of the impact of initial perturbations on tropical storm development beyond the time frame for which they are defined (Fig. 11).

One additional change is made to the initial perturbations; since there is a temperature term but no moisture term in the energy metric equation, perturbations to the basic-state temperature have the capacity to create significant changes to the relative humidity, especially over tropical oceans where the relative humidity may already be very high. The sub- or supersaturated regions created by only modifying the temperature field and leaving the moisture field constant have the capacity to inflict large changes on the model simulation through modifying moist convection (of which the dry adjoint model is unaware). To keep temperature perturbations from affecting the model primarily through the modulation of moist convection, the moisture field is modified in each case to keep the relative humidity constant. This is referred to here as a “compensating moisture” perturbation. It is noted that these moisture perturbations are not derived from sensitivity to moisture the same way that perturbations to (nondivergent) winds and temperatures are derived [Eq. (7)]—these perturbations represent a suboptimality of the perturbations that has been included only to suppress the tendency for temperature perturbations to induce significant changes early in the simulation through manipulation of the cumulus convection scheme. This is deemed a necessary sacrifice of optimality for the sake of keeping the impact of temperature perturbations relevant to the temperature sensitivities.

In addition to perturbing both winds and temperatures (with a compensating moisture perturbation), the impact of wind and temperature perturbations are examined individually to determine if one or the other is primarily responsible for influencing cyclogenesis. Since the response function is defined specifically to represent the intensity of the developing vortex at 24 h, the sensitivities of this response function are not representative of the effect perturbations would have on the TC at later times; in fact, it is possible that perturbations defined to increase the intensity of the TC at 24 h actually weaken the storm at later times. Therefore, we wish to make a careful examination of how perturbations impact the intensity of the cyclone not only at 24 h into the forecast but for the next 24 h as well.

4. Results

a. Sensitivity composites

1) Sensitivity to zonal and meridional wind

Storm-centered composites3 of low-level (0.9250 sigma) zonal flow and sensitivity to zonal and meridional flow reveal large differences between each composite group with respect to both the low-level jet and the structure of sensitivity surrounding the jet (Fig. 4). While the presence of a barotropic growth signature in the PBS composite (Figs. 4a,b) and its absence in the NBS composite (Figs. 4e,f) is clear, this is in no way surprising since this was the very criteria by which the cases were grouped. However, other differences in sensitivity structure emerge from the composites as well.

Fig. 4.
Fig. 4.

Storm-centered composites of sensitivity with respect to (left) zonal and (right) meridional flow at the initial time (shaded every 0.5 Pa s m−1, cool colors negative) and zonal flow (contoured every 1 m s−1, negative contours dashed) for (a),(b) PBS cases, (c),(d) SBS cases, and (e),(f) NBS cases.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

The zonal and meridional extent of sensitivity appears to be greatest in the PBS composite, with sensitivity contracting closer to the nascent tropical storm vortex as one progresses from the PBS to the SBS to the NBS composites. The structure of sensitivity in the PBS composite describes tilted structures implying the importance of a barotropic growth mechanism, with counterrotating vortices upstream along the shear line on the northern, cyclonic-shear side of the jet able to influence the developing tropical storm from a significant distance (recall that the composite is 1800 km on a side). The structure of the NBS composite appears to be almost entirely focused on a simple modulation of vorticity at the storm center, implying that the only significant mechanism for influencing cyclogenesis (from the linear, dry-dynamics perspective of the adjoint model) is to add or subtract vorticity from the vortex at the initial time; perturbations remote from the initial vortex have no impact on cyclogenesis in these cases. The structure of sensitivity in the SBS composite appears to be midway between these two extremes (Figs. 4c,d).

The magnitude of the sensitivity is generally much higher in PBS cases than in the other two categories; composite sensitivity to zonal and meridional flow is typically 20%–80% higher than in the SBS composite and 20%–120% higher than in the NBS composite for levels below 0.7 sigma. This implies that a smaller perturbation is capable of influencing the growth of a tropical storm vortex in these cases compared to those in the other categories. The availability of a barotropic growth mechanism by which small perturbations could quickly grow to influence cyclogenesis in the PBS category would provide a physical explanation for this phenomenon.

The basic-state jet in the PBS composite is zonally elongated with stronger shear on the northern, cyclonic-shear side over a greater zonal distance than in the other composites. Such a scenario is consistent with the notion that barotropic growth of vortices has the potential to play a larger role in PBS cases, even though the characteristics of the basic-state jet were not selected for in the grouping of cases. This is evidence that not only has our grouping method captured a sensitivity structure that appears to be consistent with barotropic growth but that these structures are related to the basic state in a way that implies a physical consistency with the barotropic growth process as well. Had cases exhibited what were thought to be barotropic growth signatures purely by chance, with no actual relationship to barotropic energy conversion, one would not expect the jet structure to be characterized by dynamically consistent features in composites of the basic state.

Cyclonic shear from the zonal flow of the jet is more intense and exists over a longer zonal extent in the PBS composite than in the NBS composite, while the SBS composite represents a middle ground (Fig. 5). In addition, conditions necessary for barotropic instability are met more easily within the environment of the PBS composite, with the meridional gradient of absolute vorticity () clearly reversing sign across the storm center. This condition likewise extends further zonally than in SBS or NBS cases.

Fig. 5.
Fig. 5.

Storm-centered composite of 0.8750-sigma zonal flow (positive values contoured in black every 1 m s−1), meridional gradient of zonal flow (negative values contoured in magenta every 1 × 10−5 s−1), and meridional gradient of absolute vorticity () (shaded every 1.0 × 10−10 m−1 s−1, cool colors negative) for (a) PBS cases, (b) SBS cases, and (c) NBS cases at the initial time. Composite south–north plot of (d) zonally averaged meridional gradient of zonal flow and (e) zonally averaged meridional gradient of absolute vorticity () for PBS cases (blue), SBS cases (green), and NBS cases (red). The “X” along the abscissa marks the location of the composite storm center, and the “J” represents the mean latitudinal position of the zonal jet maximum at this level.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

It is interesting to note that the peak magnitude of the zonal jet is slightly stronger in the SBS composite than in the PBS composite. This analysis indicates that the potential for a TC to grow through barotropic conversion of energy is highly dependent upon not only the strength of the jet but, more importantly, the magnitude of the shear on the northern, cyclonic-shear side. Therefore, a simple metric based entirely on the strength of the jet would be insufficient to determine if a tropical vortex has the potential to grow using this mechanism.

2) Sensitivity to temperature

A high degree of anticorrelation exists between sensitivity with respect to temperature and sensitivity with respect to vertical motion throughout the lower and midtroposphere (Fig. 6). The strong anticorrelation between these two fields is related to the effect of adiabatic expansion and compression on the temperature field; sensitivity with respect to upward (downward) motion is found in regions of negative (positive) sensitivity with respect to temperature because such motion promotes adiabatic cooling (warming). This relationship is clearly visible in a cross section (Fig. 6b), where positive sensitivity with respect to temperature through the core of the vortex is coincident with negative sensitivity with respect to vertical motion. The correlation coefficient is stronger than −0.7 throughout the troposphere and tapers off in the lower stratosphere (Fig. 6c).

Fig. 6.
Fig. 6.

Sensitivity with respect to initial condition temperature (shaded every 0.1 Pa K−1, cool colors negative) and sensitivity with respect to initial condition vertical motion (contoured every 2 × 10−2 Pa s m−1, negative contours dashed) for (a) the 0.7750-sigma level and (b) a west–east cross section from 0.9750 to 0.3250 sigma. (c) Profile of correlation coefficient between sensitivity with respect to initial condition temperature and sensitivity with respect to initial condition vertical motion on each sigma level.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

While this relationship is simple and relatable to adiabatic expansion and compression, it creates a problem when a attempting a dynamical analysis of sensitivity. While sensitivities do not usually speak to what may cause a perturbation to appear in a region of high sensitivity (Langland et al. 1995), vertical motion sensitivities are so well anticorrelated to temperature sensitivities that a clear cause-and-effect relationship can be established; the sensitivities with respect to vertical motion look this way precisely because such motion will cause a change in temperature. The adjoint model, being devoid of moisture physics that are necessary for the genesis of TCs, is “unaware” that subsidence through the core of the tropical vortex would be quite detrimental to the future development of the vortex. While one might otherwise expect that these sensitivities would show that warming the core of the vortex and increasing upward vertical motion would be beneficial for future development, they reveal a temperature sensitivity pattern consistent with tropical cyclogenesis but a sensitivity pattern with respect to vertical motion, directly related to the temperature sensitivity that is inconsistent.

For this reason, the vertical motion term is removed from the energy metric used to define optimal perturbations from these sensitivity gradients [Eq. (2)]. In addition, since vertical motion is accompanied by convergent and divergent flow due to mass continuity, the sensitivity with respect to the divergent component of the wind field is also removed [Eqs. (8)(9)].

b. Structure of optimal perturbations

Optimal perturbations are computed for each case as defined by Eq. (2) and binned into each category. The dynamical sensitivity analysis performed on composites (section 4a, above) allows speculation about how the structure of optimal perturbations differs between PBS and NBS cases. In general, we anticipate that since PBS cases are cases in which a barotropic energy conversion mechanism is available to grow perturbations and influence cyclogenesis, optimal perturbations will be focused more toward the jet, investing a larger percentage of perturbation energy at jet level, compared to NBS cases, which do not have access to this growth mechanism. The enhanced magnitude of sensitivities in PBS cases also implies that for the chosen target ( in 24 h) PBS cases will require less energy overall to affect that change than will NBS cases. The first of these predictions involves the vertical structure of perturbations without regard to magnitude, while the second involves only the magnitude of perturbations.

Mean profiles of total energy for each bin confirm these speculative differences between PBS and NBS optimal perturbations (Fig. 7). Normalized profiles show that peak energy investment occurs within the jet level for all categories, but PBS cases invest far more at these levels than NBS cases (Fig. 7a), with differences surpassing a one-sided Student’s t test at 95% confidence for all levels between 0.9750–0.8250 sigma. The nonnormalized profiles show the total amount of energy invested at every level in PBS cases is smaller than in NBS cases (Fig. 7b), surpassing a two-sided Student’s t test at 95% confidence at all levels. Unsurprisingly, the mean SBS profile is midway between the PBS and NBS profiles in both plots. The characteristically large magnitude of NBS (optimal) perturbations yields larger growth rates of these perturbations as a byproduct of their size (see below). The structure of perturbations follows the sensitivity structure, with PBS cases describing temperature and vorticity perturbations in counterrotating vortices along the shear line and NBS cases creating a stronger, warmer core, while SBS cases again express the middle ground between these extremes (Figs. 7c–e).

Fig. 7.
Fig. 7.

Vertical structure of mean optimal perturbations for each category. Mean profiles of (a) percent of total energy added to each sigma level and (b) amount of energy added to each sigma level (PBS = blue, SBS = green, NBS = red). The gray line corresponds to a profile of zonal-average flow through the core of the composite jet amongst all cases. Magenta line along y axis highlights levels where differences between PBS and NBS profiles surpass a Student’s t test at 95% confidence. West–east cross section of perturbation temperature (shaded every 0.1 K, cool colors negative) and perturbation vorticity (contoured every 1 × 10−5 s−1, negative contours dashed) for (c) PBS mean, (d) SBS mean, and (e) NBS mean.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

Dividing these profiles into their kinetic energy and available potential energy components shows that the majority of perturbation energy in the lower troposphere (65%–80%) is composed of kinetic energy (Fig. 8a). In addition, temperature perturbations (especially in the PBS composite) display the same upshear-tilted structures observed in the wind perturbations (Fig. 8b), implying a connection to the barotropic growth processes. This can be understood by recognizing that a balanced wind perturbation can impose a greater influence on the response function than an unbalanced one, because some energy in the unbalanced perturbation is lost to adjustment. Thus, a temperature perturbation that keeps a wind perturbation in balance is capable of influencing the response function through the barotropic growth process. Indeed, when only the temperature is perturbed (with compensating moisture), very quickly the temperature perturbations evolve to produce wind perturbations with coherent features very similar to perturbations evolved from a wind-only initial perturbation (Figs. 8c,d).

Fig. 8.
Fig. 8.

(a) Composite profiles of summed perturbation kinetic (solid lines) and available potential (dashed lines) energy on each sigma level for the PBS composite (blue), SBS composite (green), and NBS composite (red). (b) Storm-centered composite of 0.8750-sigma optimal temperature perturbations for PBS cases (shaded every 0.05 K, cool colors negative), composite vorticity (magenta contours every 1.0 × 10−5 s−1, positive values only), and composite zonal flow (black contours every 1 m s−1, positive values only). (c) Cross section SW–NE through Tropical Storm Alma (2008) forecast at 9 h into the simulation. Relative vorticity (shaded every 4 × 10−5 s−1, positive values only), perturbation zonal flow from the (u, υ)-perturbation experiment (black contours every 3 m s−1, negative contours dashed), and perturbation zonal flow from the (T, q)-perturbation experiment (magenta contours every 3 m s−1, negative contours dashed). (d) As in (c), except perturbation meridional flow is contoured.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

The potential for a perturbation to grow barotropically can be assessed from an energetics perspective by evaluating the deformation of a perturbation in a background field with any general local strain [Mak and Cai 1989, their Eq. (29)]. Here, not only is the orientation of a perturbation relative to the background shearing deformation considered but also the orientation of a perturbation relative to the background stretching deformation. This formulation provides an estimate of the local (barotropic) energy generation rate. The formulation from Eq. (29) of Mak and Cai (1989) is considered here because it can be used to derive the energy generation rate due to basic-state shearing and basic-state stretching individually. This is of particular value when discussing the influence of a low-level zonal jet with significant zonal-shear flow on the energetics of a TC vortex.

Composites of the local energy generation rate due to basic-state shearing and stretching deformation show that the shearing deformation is responsible for the bulk of the energy generation in all three composites (Fig. 9). Most of the energy generation in the PBS composite is occurring in the high-shear region upstream of the nascent vortex (Figs. 9a,b), while maximum energy generation appears in the northeast quadrant of the vortex in the NBS composite (Figs. 9e,f). The SBS composite is midway between these extremes (Figs. 9c,d).

Fig. 9.
Fig. 9.

Storm-centered composites of localized energy generation rate of wind perturbations from (left) basic-state shearing deformation and (right) basic-state stretching deformation at the 0.8750-sigma level. Composites are shown for (a),(b) the PBS category, (c),(d) the SBS category, and (e),(f) the NBS category. Energy generation rate is shaded every 4 J Kg−1 s−1 (cool colors negative). Basic-state vorticity is contoured in magenta every 1 × 10−5 s−1 (positive values only), and basic-state zonal flow is contoured in black every 1 m s−1 (positive values only).

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

When comparing the size of growth rates, it is important to recognize that perturbations in the NBS cases are typically larger (see Fig. 7b), leading to faster growth rates all else being equal. The localized energy growth rate following Mak and Cai (1989) can be described as
e16
where the u and the υ represent the perturbation values and U and V represent the basic-state values. The terms and are dependent upon the size of the perturbation. In two cases with differing sensitivity but similar basic-state deformation, the case with lower sensitivity requires larger (optimal) perturbations, yielding larger growth rates simply by virtue of their larger size. To compensate for this fact, one can compute the localized energy growth rate normalized by the initial perturbation energy to yield a growth rate per J kg−1 of initial perturbation energy. Profiles of this normalized quantity reveal that the PBS composite has the most efficiently growing perturbations (Fig. 10), especially at jet level.
Fig. 10.
Fig. 10.

Composite profiles of summed energy growth rate normalized by summed initial perturbation energy by sigma level for (blue) PBS cases, (green) SBS cases, and (red) NBS cases.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

c. Impact of optimal perturbations

Optimal perturbations are introduced into the model initial conditions, and another 48-h simulation is produced for each case. It is desirable to determine if sensitivities of intensity defined for the 24-h model state promote continued intensification of the cyclone at later times or if perhaps these perturbations represent a highly nonbalanced solution that increases intensity at verification of the response function and then quickly become a neutral or even negative influence on intensity. Hence, the control and perturbed simulations are run for the 24 h constituting the trajectory over which the adjoint model is linearized as well as the 24 h that follow. In addition, the perturbed runs include a simulation with perturbations to winds and temperatures (with compensating moisture perturbations to keep relative humidity constant), a simulation where only the wind components are perturbed, and a simulation where only the temperature (with compensating moisture) is perturbed. This is done to determine if one of these components dominates the impact on cyclogenesis.

Optimal perturbations appear to intensify all three categories of tropical storm vortices both at 24 h and 48 h into the simulation (Fig. 11). Intensification at 24 h appears to be slightly less than the expectation of 1.5 hPa in the ~250-km box at storm center and is likely due to the effects of nonlinear, full moisture physics that are missing in the tangent linear and adjoint models. Reduction in sea level pressure at 48 h is greater than at 24 h in all three composites, implying that optimal perturbations intensify the tropical vortex in relevant ways that can have significant impacts on longer-term forecasts.

Fig. 11.
Fig. 11.

Storm-centered composites of sea level pressure (contoured every 1 hPa) and difference in sea level pressure between perturbed and unperturbed simulation (shaded every 0.5 hPa, cool colors negative) at (left) 24 h and (right) 48 h for (a),(b) PBS cases, (c),(d) SBS cases, and (e),(f) NBS cases.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

Specifically examining the response function, defined as the change in perturbation pressure in the lowest sigma level within a 250 km × 250 km box centered on the final-time position of the vortex in the unperturbed simulation, several features can be observed when comparing (u, υ) perturbations, (T, q) perturbations, and (u, υ, T, q) perturbations separately (Fig. 12). First, the intensification of the vortex is coincident with a shift in the final-time position of the vortex in the NW–SE direction, as evidenced by a dipole in the change in perturbation pressure. Second, the impact of (u, υ, T, q) perturbations on R (Fig. 12a) appears to be more similar to the impact of (u, υ) perturbations (Fig. 12b) than (T, q) perturbations (Fig. 12c). This is further evidence indicating that the role of temperature perturbations is, in part, to enhance the impact of the wind perturbations through establishing balance to the wind perturbations early in the simulation.

Fig. 12.
Fig. 12.

Response function box-centered composite across all cases of change in perturbation pressure at 24 h into the simulation for (a) (u, υ, T, q)-perturbation test, (b) (u, υ)-perturbation test, and (c) (T, q)-perturbation test. Difference in perturbation pressure between perturbed and unperturbed simulations is shaded every 20 Pa (cool colors negative). The magenta box outlines the confines of the response function box. (d) Average observed change in response function across all cases for (left)–(right) (u, υ)-perturbation test, (T, q)-perturbation test, the sum of these functions, the (u, υ, T, q)-perturbation test, and the desired change in the response function defining optimal perturbations. Values are given in hPa.

Citation: Journal of the Atmospheric Sciences 72, 3; 10.1175/JAS-D-14-0053.1

As an ad hoc measure of linearity, one can compute the change in R from (u, υ) perturbations and (T, q) perturbations and sum them, and then compare that value to the change in R from (u, υ, T, q) perturbations (Fig. 12d). These perturbations are all evolved within the nonlinear NWP model with full moisture physics; it can be assumed that the differences that arise between these two values for are due to nonlinear evolution of perturbations when wind and temperature–moisture perturbations are added versus adding each type of perturbation separately. Examining the values of computed this way reveals that they are quantitatively similar to each other within a tolerance of ±3%. This is strong evidence that nonlinear processes do not play a major role in evolving these perturbations.

One can also compare the observed to the value estimated by the adjoint model: . Differences between these two values must come from one or more of the following sources: 1) nonlinear evolution of perturbations in the model, 2) physics that exist within the NWP model but not the adjoint model (including moist convection and radiative cooling), and 3) the inclusion of compensating moisture that was not accounted for in the original calculation of . Comparing these values shows that the adjoint model typically overestimates the change in R for these perturbations by about 37%. Since the previous test showed that the effect on nonlinearity on the evolution of perturbations was likely small, this deficit must come from the remaining sources. While the (linear, adiabatic) adjoint model anticipates that wind perturbations should be responsible for 63.3% of the total impact on intensity, with temperature perturbations responsible for the remaining 36.7%, the perturbations to the nonlinear, full-physics model essentially reverse these values—63.9% of the impact comes from the temperature (and compensating moisture)-only perturbations, while only 36.1% comes from the wind-only perturbations. This provides further evidence that the deficit described above can be traced back to the inclusion of moist convection and moisture perturbations in the nonlinear model.

5. Conclusions

Though it has been inferred that TCs in the eastern Pacific can take advantage of energy in the barotropic shear of their environment and grow barotropically, this potential for barotropic growth has since been difficult to identify; there is very little in the analysis fields themselves that can be used to show conclusively that a TC is inclined to grow barotropically in the environment provided precursor disturbances are oriented in a particular way, and the barotropic growth mechanism is not the sole cause of TC genesis in the eastern Pacific basin.

Using the adjoint of an NWP model, it has been shown that sensitivities of (nascent tropical cyclone) vortex intensity often exhibit structures along the low-level westerly jet that tilt upshear, indicating the potential for barotropic growth. A survey of nearly 50 cases of tropical storm genesis in the eastern Pacific from 2004 to 2010 shows that roughly 80% of them exhibit some kind of barotropic-like structure, with sensitivity to barotropic energy conversion being a priority in 35% of total cases. Composites of these cases into bins based on the relative strength of this signature reveal that cases in which barotropic growth plays a primary role are typified by a strong low-level jet with enhanced meridional shear across a wide zonal extent and sensitivity gradients with respect to wind perturbations are of a higher magnitude. Optimal perturbations to increase intensity were found to be collocated with the low-level jet, with significantly more total (dry) energy contributed at the level of the jet in PBS cases than NBS cases. Using the framework of Mak and Cai (1989) to define barotropic energy growth rates, it was found that the largest growth rates in the PBS composite reside along the low-level westerly jet, while in the NBS composite the largest growth rates are found within the northeast quadrant of the tropical storm vortex. Normalized by total initial perturbation (kinetic) energy, the perturbations in the PBS composite are found to achieve the most efficient growth.

Relatively little perturbation energy is added to the initial conditions of the PBS cases, resulting in a deepening at both 24 and 48 h, while in the NBS cases substantially more energy was required to achieve the same result. It was also found that the majority of the observed deepening can be achieved by perturbing the temperature field alone (with compensating moisture). Disagreement between adjoint estimates and the behavior of perturbations in nonlinear, full-physics simulations is likely due to the lack of moist convection in the adjoint model and the inclusion of nonoptimal (but necessary) moisture perturbations unaccounted for in the adjoint estimate.

Adjoint-derived sensitivity gradients provide a wealth of dynamical information about the evolution of specific aspects of the model forecast that is otherwise difficult or impractical to produce. Even in the case of TC genesis, where nonlinearity and the importance of moisture physics would otherwise place adjoint models at a distinct disadvantage, a sufficiently well-crafted study can make use of this sensitivity information to provide direct evidence of a potentiality only hinted at by more traditional methods. When properly exercised as a dynamical tool, the adjoint model provides new insights and creates new directions for dynamical research.

It is important to recognize that this simulation was performed with an adjoint model employing adiabatic physics. The effect of moist physics was not considered in this study for the sake of maintaining realistic results in the adjoint model simulations. Sophisticated adjoint models now exist that can explicitly handle the physics of parameterized moist convection, including the WRF, COAMPS (e.g., Doyle et al. 2012), and GEOS-5 (Holdaway and Errico 2014), and the inclusion of these moisture physics and their interaction with the barotropic growth mechanism in east Pacific tropical cyclogenesis is an area of further study. The results shown here omit the impact of moisture where moisture likely plays a significant role, and previous results using a similar technique with a moist adjoint have found moisture perturbations to be dominant (Doyle et al. 2012).

Results of this study prompt further questions about east Pacific TC genesis. What kinds of precursor disturbances are most likely to take advantage of this potential growth mechanism? What kinds of errors in initial conditions cause the most egregious forecast errors of TC genesis and intensity based on these sensitivities, and how can those errors be combated most effectively? Does this growth mechanism operate in other basins? How might the relative importance of barotropic growth processes help delineate between more predictable and less predictable TCs? Adjoint models may be used alongside more traditional methods in the future to answer these questions.

Acknowledgments

The first author is supported by the National Science Foundation under Grant 0529343 and the Office of Naval Research under Grant N000141110609. The author would like to thank Dr. Michael Morgan at the University of Wisconsin–Madison for his guidance and insight.

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1

The data for this study are from the Research Data Archive (RDA) maintained by the Computational and Information Systems Laboratory (CISL) at the National Center for Atmospheric Research (NCAR). NCAR is sponsored by the National Science Foundation (NSF). The original data are available from the RDA (http://rda.ucar.edu/) in dataset number ds083.2.

2

The original data are available from the RDA (http://rda.ucar.edu/) in dataset number ds335.0.

3

A few cases are omitted from the storm-centered composites because of technical problems (e.g., there appears to be no initial disturbance to define a storm center, perturbations create a CFL criteria failure, etc.). Composites at initial time and 24 h omit five cases (one SBS, one NBS, and three PBS), and composites at 48 h omit two additional PBS cases that make landfall before 48 h.

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