Abstract

The steering of a tropical cyclone (TC) vortex is commonly understood as the advection of the TC vortex by an “environmental wind.” In past studies, the environmental steering wind vector has been defined by the horizontal and vertical averaging of the horizontal winds in a box centered on the TC. The components of this environmental steering have been proposed as response functions to derive adjoint-derived sensitivities of TC zonal and meridional steering. The appropriateness of these response functions in adjoint sensitivity studies of TC steering is tested using a two-dimensional barotropic model and its adjoint for a 24-h forecast. It is found that these response functions do not produce sensitivities to TC steering because perturbations to the model initial conditions that change the final-time location of the TC also change the response functions in ways that have nothing to do with the steering of the TC at model verification.

An alternate response function is proposed wherein the environmental steering vector is defined as the wind averaged over the response function box attributed to vorticity outside of that box. By redefining the response functions for the zonal and meridional steering as components of this environmental steering vector, the effect of small changes to the final-time location of the TC is removed, and the resultant sensitivity gradients can be shown to truly represent the sensitivity of TC steering to perturbations of the model forecast state.

1. Introduction

An adjoint sensitivity study involves evaluating the change in a specific aspect (called a response function R) of a model forecast state [xf = x(tf)] resulting from arbitrary changes in any of the model control variables at the initial (t = 0) or forecast times (t = τ). Such a study calculates the gradient of the response function ∂R/∂xτ with respect to the model control variables represented by a state vector (xτ).1 An adjoint model is the most efficient tool for evaluating these forecast sensitivities (Errico 1997).

Conspicuously absent from the many prior applications of adjoint-based sensitivity analysis to tropical and extratropical cyclone issues are synoptic and dynamical interpretations of these sensitivities. When dynamical interpretation of sensitivities is provided, often what is offered is merely the observation that the distribution of sensitivities (Vukicevic and Raeder 1995; Wu et al. 2007) or singular vectors (Peng and Reynolds 2006; Chen et al. 2009) is coincident with a synoptic feature. Coincidence of adjoint sensitivities with a synoptic feature alone is insufficient to attribute dynamical significance to the feature (Langland et al. 1995). Langland et al. (1995) demonstrate that the sensitivity fields calculated in their study of an idealized cyclogenesis event in a time-evolving, nonzonal flow were very similar to the same calculation performed for a steady, zonal basic state. While adjoint sensitivities do not provide information regarding whether particular physical processes actually occurred within a basic state (control) forecast trajectory, these sensitivities do provide information concerning the effect of possible perturbations to the basic state. It is this particular characteristic of adjoint sensitivities that makes them a potentially powerful tool in synoptic case studies (Langland and Errico 1996).

In order for the results of an adjoint sensitivity calculation to have meaning, thorough testing and interpretation is required (Errico and Vukicevic 1992). Such rigorous dynamical interpretation and testing of adjoint-derived forecast sensitivities provides a means of characterizing what the sensitivity fields represent and evaluates how well the response function associated with the sensitivity gradients truly measures its intended forecast aspect. In this study, a specific set of response functions, designed to measure instantaneous tropical cyclone (TC) steering, is considered. It is demonstrated that while the response functions chosen are appropriate for diagnosing steering, the sensitivities calculated with these response functions do not provide insight into the sensitivity of TC steering. A solution to this problem is offered and tested.

Wu et al. (2007) presents an objective targeting strategy whereby a response function is defined to represent the zonal or meridional steering of a TC at model verification:

 
formula
 
formula

where R1 and R2 represent, respectively, the horizontally and vertically averaged zonal and meridional wind in a horizontal domain D and bounded between 850 and 300 hPa (indexed by k). Here, D, or the “response function box,” is defined as the set of all grid points (indexed zonally by i and meridionally by j) in a box 600 km on a side, centered on the model verification position of the TC. In this way, the response function R1 (R2) is the environmental zonal (meridional) wind steering the TC at verification (Chan and Gray 1982). The sensitivity gradients can be combined into a vector

 
formula

that represents the vector change in steering of the TC given a unit perturbation to a model state vector x. Wu et al. (2007) call these vectors adjoint-derived sensitivity steering vectors (ADSSVs) and use the vectors in an attempt to understand the influence of dropsondes deployed in a Dropwindsonde Observations for Typhoon Surveillance near the Taiwan Region (DOTSTAR) targeted observation field campaign (Wu et al. 2005) to initialize model forecasts of Typhoon Mindulle (2004). Wu et al. (2007) suggest that because these dropsondes provided data in regions of low sensitivity their assimilation would not have significantly improved track forecasts for Mindulle.

While care has been taken to show that sensitivities using these TC steering response functions are coincident with synoptic features likely important to the steering of a modeled TC (Wu et al. 2009), this coincidence is insufficient to identify those synoptic features as important. Further, no studies exist that test the validity of these response functions by perturbing the initial conditions of the model in regions of high sensitivity and determining exactly how those perturbations impact the response function. Because of the lack of any rigorous testing of the appropriateness of these response functions and a lack of dynamical interpretation of the sensitivity fields, it is unclear whether these sensitivity gradients appropriately describe sensitivity to TC steering.

Using a 2D, barotropic, nondivergent, inviscid model on an f plane (see section 2), it is shown in this study that the response functions previously used to define TC steering (see Wu et al. 2007, 2009) are inappropriate for diagnosing TC steering sensitivity. A dynamical interpretation of the sensitivity gradients for R1 and R2 in the simplified model provides an explanation of the problem as well as the solution. Section 2 provides a description of the model and the idealized case used in the study. A dynamical interpretation of the sensitivities is provided in section 3, along with an analysis of perturbations introduced into the model to verify the interpretation. In section 4, a solution to this problem is tested and verified by redefining the response functions used by Wu et al. (2007) to describe TC steering. Future work using these tools is described in section 5.

2. Model study

a. The model

The simplified numerical model used is based on the two-dimensional, nondivergent, inviscid, barotropic vorticity equation on an f plane, for which relative vorticity is conserved:

 
formula

where ζ is the relative vorticity and V is the horizontal vector wind field with zonal and meridional components u and υ, respectively. A streamfunction ψ, which is related to the relative vorticity by ∇2ψ = ζ, is diagnosed from the vorticity distribution using successive overrelaxation. The nondivergent wind field is calculated from the streamfunction:

 
formula
 
formula

The nonlinear model is solved on a centered-difference, discretized f plane over a closed domain 5235 km across with a grid spacing of 15 km. Homogeneous lateral boundary conditions are imposed on the streamfunction. The model is integrated forward in time using a forward Euler scheme with a 30-s time step.

b. The tangent-linear and adjoint models2

Linearization of (4) about a time-evolving forecast trajectory calculated from (4) yields the tangent linear model:

 
formula

where ζ and V are the basic state relative vorticity and wind vector V = (u, υ), respectively, and ζ′ and V′ are the perturbation relative vorticity and wind vector V′ = (u′, υ′), respectively. The tangent linear model is also solved using a centered in space, forward Euler in time scheme. The adjoint model is developed at the coding level as the line-by-line transpose of the tangent linear model. Tests of the validity and accuracy of the tangent linear and adjoint codes are performed in sections 3b, 4c, and 4d. Because the dynamics of the model are expressed in terms of the distribution of relative vorticity, the output of the adjoint model is sensitivity with respect to vorticity.

c. Idealized case

To aid dynamical interpretation of adjoint-derived sensitivities of TC steering, the simplest possible idealization was chosen: a TC embedded in a quiescent environment. The nonlinear model was initialized with a Gaussian distribution of vorticity [hereafter referred to as the basic-state vortex (BSV)] maximized at the center of the model domain (Fig. 1). This vortex is the “TC” in the model. The model is run for 24 h. Since the only wind in the model domain is symmetric about the BSV center, the state of the model is unchanged for the full 24-h integration. This 24-h model trajectory defines the basic state about which the adjoint model is linearized.

Fig. 1.

(a),(c) Basic state vorticity contoured every 2.0 × 10−5 s−1 and sensitivity of R1 with respect to vorticity shaded every 5.0 × 104 m at (a) model verification and (c) model initialization. (b),(d) As in (a),(c), respectively, but for sensitivity of RE1 with respect to vorticity. (e) Perturbation vorticity shaded every 5.0 × 10−6 s−1 and perturbation winds at model verification after optimal perturbations to increase R1 were introduced at model initialization. (f) Perturbation environmental vorticity shaded every 5.0 × 10−7 s−1 and perturbation environmental winds at model verification after optimal perturbations to increase RE1 were introduced at model initialization. Vectors in (f) are an order of magnitude smaller than vectors in (e). The box at the center of the plot corresponds to the boundaries of the response function box. Note that the wavy patterns in sensitivity in (d) are due to the zeroth-order discontinuity in initialized sensitivity [see (b)] and do not negatively impact the results.

Fig. 1.

(a),(c) Basic state vorticity contoured every 2.0 × 10−5 s−1 and sensitivity of R1 with respect to vorticity shaded every 5.0 × 104 m at (a) model verification and (c) model initialization. (b),(d) As in (a),(c), respectively, but for sensitivity of RE1 with respect to vorticity. (e) Perturbation vorticity shaded every 5.0 × 10−6 s−1 and perturbation winds at model verification after optimal perturbations to increase R1 were introduced at model initialization. (f) Perturbation environmental vorticity shaded every 5.0 × 10−7 s−1 and perturbation environmental winds at model verification after optimal perturbations to increase RE1 were introduced at model initialization. Vectors in (f) are an order of magnitude smaller than vectors in (e). The box at the center of the plot corresponds to the boundaries of the response function box. Note that the wavy patterns in sensitivity in (d) are due to the zeroth-order discontinuity in initialized sensitivity [see (b)] and do not negatively impact the results.

The response function is defined as the average zonal flow in a 1200 km × 1200 km box centered on the BSV at the end of the 24-h model integration:

 
formula

For brevity, the discussion to follow focuses on response functions for zonal steering.

3. Description and interpretation of ∂R1/∂ζ

a. Dynamics

Before describing the sensitivities of R1 to vorticity, we consider what perturbations to the model forecast trajectory would change R1, the average zonal flow in the response function box at forecast hour 24 given the dynamical constraints of the model. The only way to increase R1 at 24 h is for positive (negative) relative vorticity perturbations to be found north (south) of the center latitude of the box at that time. Within the context of the linearized model, there are only two processes that can create this state: 1) the basic state flow advects positive (negative) perturbation vorticity north (south) of the BSV position or 2) the creation of vorticity perturbations associated with the northward advection of the BSV (translation of the TC) by winds associated with vorticity perturbations external to the BSV. The former effect, the basic-state advection of perturbation vorticity [second term in the linearized model (7)], is the instantaneous steering effect we seek to capture using the adjoint sensitivity diagnosis. The latter effect, an increase in R1 due to a northward displacement of the BSV [manifest as an asymmetric distribution of perturbation vorticity in the box, third term in (7)] is not a zonal steering effect. Furthermore, this latter effect represents the integrated response of the imposed vorticity perturbations west and east of the longitude of the BSV advecting that vortex northward rather than the instantaneous response associated with the north–south perturbation vortex dipole.

b. Adjoint sensitivity gradients for R1

The adjoint model is initialized with the distribution of ∂R1/∂ζ at the end of the 24-h model integration (Fig. 1a).3 These sensitivities describe how a perturbation to vorticity would instantaneously change the response function. Sensitivities at this time are positive (negative) to the north (south) of the BSV (Fig. 1a). This distribution of sensitivities implies that placing positive (negative) vorticity perturbations north (south) of the response function box increases R1. Such perturbations correspond to an increase in the zonal flow that would steer the BSV (the TC) eastward at this time.

Integrated backward in time 24 h to the model initialization time, ∂R1/∂ζ has two important features (Fig. 1c). First, the maximum positive (negative) sensitivity appears northeast (southwest) of the BSV center. Second, positive (negative) sensitivity now appears west (east) of the BSV center. Each of these two features can be identified with processes in (7): the advection of perturbation vorticity by the basic state wind field (the instantaneous steering effect) and the advection of the basic state vorticity by the perturbation wind field (the primary cyclone displacement effect).

c. Interpretation

1) Instantaneous steering effect

The sensitivity of R1 with respect to vorticity at 24 h is a north–south dipole of positive and negative vorticity centered on the BSV. Because the absolute (and relative) vorticity is conserved on an f plane in the model, at forecast hours prior to 24 h these vorticity perturbations must be located upstream of their final-time locations (Fig. 2a). How far upstream the vorticity perturbations must be located to maximize their influence on increasing R1 is dependent on the length of the adjoint model backward integration, the distance the vorticity perturbation is from the center of the box, and the intensity of the BSV.

Fig. 2.

(a) Schematic configuration of vorticity perturbations (+ for positive perturbation, − for negative) at initial forward model time placed in regions of maximum sensitivity shown in Fig. 1c; (b) subsequent configuration of vorticity perturbations 24 h later that contribute to the instantaneous zonal flow within the response function box; (c) configuration of vorticity perturbations contributing to a northeastward “primary cyclone displacement” effect at the forward model initial time; and (d) subsequent configuration of the BSV and vorticity perturbations 24 h after the time shown in (c). Note that in (d) the BSV has moved north-northeast of its initial location. Tropical cyclone symbol denotes location of BSV. Black filled arrows indicate flow associated with BSV. Gray filled arrows show direction of flow at vortex center attributed to perturbation vorticity. Shading indicates approximate regions of maximum sensitivity for R1 that would produce the steering effect [in (a) and (b)] and the PCD effect [in (c)].

Fig. 2.

(a) Schematic configuration of vorticity perturbations (+ for positive perturbation, − for negative) at initial forward model time placed in regions of maximum sensitivity shown in Fig. 1c; (b) subsequent configuration of vorticity perturbations 24 h later that contribute to the instantaneous zonal flow within the response function box; (c) configuration of vorticity perturbations contributing to a northeastward “primary cyclone displacement” effect at the forward model initial time; and (d) subsequent configuration of the BSV and vorticity perturbations 24 h after the time shown in (c). Note that in (d) the BSV has moved north-northeast of its initial location. Tropical cyclone symbol denotes location of BSV. Black filled arrows indicate flow associated with BSV. Gray filled arrows show direction of flow at vortex center attributed to perturbation vorticity. Shading indicates approximate regions of maximum sensitivity for R1 that would produce the steering effect [in (a) and (b)] and the PCD effect [in (c)].

Perturbation vorticity northeast and southwest of the BSV at the forecast hour 0 will be advected by the basic state wind field (Fig. 2a). At the end of the 24-h model integration, this perturbation vorticity will be directly north and south of the BSV center (Fig. 2b). Therefore, if positive (negative) perturbation vorticity were introduced northeast (southwest) of the BSV at model initialization, the dipole of positive and negative perturbation vorticity would be oriented north–south across the response function box by the end of the model integration. This perturbation vorticity would be associated with a positive zonal perturbation flow in the response function box, increasing R1.

2) Primary cyclone displacement effect

Unlike the instantaneous steering effect, the distribution of perturbation vorticity which would create a northward displacement of the BSV is less obvious because of the dual (possibly competing) effects of the imposed vortices: the vortices will not only perturb the location of the BSV but will also directly contribute to the zonal flow in the box at the time the response function is defined.

In order for perturbation vorticity to advect the BSV northward, the positive (negative) perturbations must be located northwest (southeast) of the BSV at some time over the 24-h period. In this configuration, the perturbation flow would be from the south over the BSV. Positive (negative) perturbation vorticity placed initially to the west (east) of the BSV would be associated with a southerly perturbation flow, advecting the BSV to the northeast (Fig. 2c). By the end of the 24-h model integration, the BSV is displaced northeast of the center of the response function box (Fig. 2d). The symmetric circulation of the BSV contributes positively to the average zonal flow in the response function box, positively influencing R1.

The dynamical interpretation of the adjoint-derived sensitivity gradients makes the problem with the R1 clear: the process described by the basic state advection of perturbation vorticity actually impacts the steering of the BSV. Sensitivities associated with this process can legitimately be called sensitivities of TC steering. The sensitivities associated with perturbation advection of basic state vorticity are of another sort. Perturbations to the model initial conditions in regions where these sensitivities are large result in the BSV being advected north of its final-time location in the control forecast and therefore contribute to R1 in a way that has nothing to do with the zonal steering of the BSV. We call this the primary cyclone displacement (PCD) effect.

While it would be convenient to simply move the averaging box to center it over the new vortex center and thus remove the effect of the displaced cyclone, the response function is chosen as the average zonal wind in a box over a specific geographic region. While the chosen region may be centered directly on the vortex in the basic state, and therefore represent the steering of a modeled TC in the basic state, there is nothing preventing vorticity perturbations from advecting the vortex out of the middle of the box. Any comparison, qualitative or quantitative, between these sensitivities and the steering of the modeled TC must keep the averaging box in the same place. The response function box cannot be arbitrarily moved based on a posteriori information to accommodate the perturbation advection of the BSV.

d. Perturbation test for validation of dynamical interpretation

A test can be performed to validate the above dynamical interpretation for ∂R1/∂ζ. Initial vorticity perturbations, designed to increase R1, may be derived by scaling the sensitivity to the initial distribution of vorticity, :

 
formula

Here, is the absolute value of the maximum sensitivity in the model domain at forecast time 0 h. An initial perturbation vorticity distribution with maximum perturbation vorticity 2.0 × 10−6 s−1 is created by choosing S = 2.0 × 10−6. In this way, the perturbation vorticity introduced at model initialization has the same structure as ∂R1/∂ζt=0h (Fig. 1c).

The model is integrated forward 24 h with the perturbed initial conditions. At 24 h, perturbation vorticity and winds—defined as the difference between the perturbed and control (unperturbed) forecast values of vorticity and wind, respectively—reveal a dipole of perturbation vorticity in the response function box, indicating a northeast translation of the BSV by the perturbation wind field (Fig. 1e). A significant portion4 of the perturbation flow in the response function box is attributed to the vorticity perturbations ascribed to the translation of the BSV. This result confirms the dynamical interpretation for the fault with ∂R1/∂ζ, namely, that the assumption that the BSV would remain in the center of the response function box is violated, and some contributions to the change in R1 have nothing to do with the instantaneous zonal steering of the BSV at 24 h. In fact, it appears that for this case the PCD effect is larger than the instantaneous steering effect.

We can also perform a test to validate the adjoint model’s assumption of linearity for this case. The total change in the response function between the perturbed and control runs can be directly calculated by evaluating R1 for each run and calculating the difference:

 
formula

Assuming linearity, this value can be approximated by evaluating the inner product of the sensitivity field at model initialization ∂R1/∂ζt=0h with the perturbation to initial condition vorticity ζt=0h:

 
formula

For the case under consideration, ΔR1=0.922 m s−1, while δR1 = 0.919 m s−1. The adjoint model was able to account for 99.7% of the change in R1, indicating that the perturbations evolved linearly in the model, and the adjoint-derived sensitivity gradients are valid.

As Fig. 1e shows, the change in R1 is largely the result of a slight northward displacement of the final-time location of the BSV within the response function box. This does not correspond to an increase in the zonal steering of the BSV at this time. The PCD effect does not appear to be dependent on the physical size of the BSV (not shown).

The strength of the PCD effect is partially dependent on the size of the response function box. The PCD effect is manifest as a dipole of perturbation vorticity in the center of the box with a concomitant perturbation flow pattern described by that vorticity. While a larger response function box would do nothing to reduce the size of this dipole, the contribution of the PCD effect on the average perturbation flow would be smaller for a large response function box than for a small box. However, this cannot serve as a solution to this problem. The PCD effect exists regardless of the size of the response function box. And the box cannot be too large, since the response function is intended to describe the environmental flow in the vicinity of the TC.

In the following section, a new response function is introduced to provide a solution to this problem and is tested in the same manner as R1.

4. Proposed solution

a. Environmental steering response function

It has been shown that the response function used to describe TC steering is fundamentally flawed. Perturbations to the final-time location of the TC within the response function box allow for the TC’s own symmetric circulation to contribute to R1 (and R2) in a way that has nothing to do with the instantaneous steering of the TC. Dynamical interpretation of ∂R1/∂ζ shows that the PCD effect is manifest as positive (negative) sensitivity with respect to vorticity appearing west (east) of the BSV.

Here, we propose new response functions to define the zonal and meridional steering of a TC, such that the PCD effect is removed from the sensitivity gradients associated with the new response functions. For a given domain, one can define the “hurricane advection flow” as the balanced flow at the TC center attributed to the potential vorticity (PV) within that domain after the PV of the TC has been removed (Wu and Emanuel 1995).

For two-dimensional, nondivergent, barotropic flow, the relevant potential vorticity is the absolute vorticity. We designate the vorticity within the response function box D as vorticity of the BSV, and all of the vorticity outside of the response function box as vorticity of the environment. The partitioning of the full domain vorticity into vorticity of the TC (vorticity in box) and vorticity of the environment is accomplished by the use of a linear environmental projection operator E:

 
formula

where ζi,j represents the gridpoint representation of the vorticity.

Once the vorticity of the TC is removed, the vorticity of the environment can be inverted to recover the “environmental” wind, : = Q−1(Eζ) = k × [∇−2(Eζ)] = ũi + υ̃j, where Q is an operator that calculates the two-dimensional, nondivergent wind field from the vorticity distribution and ∇−2 represents the successive overrelaxation inversion operator that calculates streamfunction from vorticity.

We define a new set of response functions to describe the average environmental zonal and meridional wind in the vicinity of the TC:

 
formula
 
formula

The averaging of the environmental wind (ũ, υ̃) is performed in the same manner as the averaging of the full wind in R1 and R2. Figure 3a is a flowchart describing the procedure used to calculate RE1 and RE2 given the model state as expressed in terms of vorticity. The adjoint of this procedure, shown in Fig. 3b, is used to derive the sensitivity gradients that initialize the adjoint model.

Fig. 3.

(a) Flowchart describing how the response functions RE1 and RE2 are calculated from final time forecast vorticity ζf . Here E is the environmental projection operator. The operator that inverts vorticity and calculates the horizontal wind attributed to that vorticity is Q−1. The horizontal wind field attributed to vorticity outside of the response function box is ũf , υ̃f . (b) Flowchart describing how the initial conditions to the adjoint model, ∂RE1/∂ζf and ∂RE1/∂ζf , are calculated from ∂RE1/∂ũf and ∂RE1/∂f . Operators E* and (Q−1)* are the adjoints of E and Q−1, respectively. Note that for the zonal environmental wind response function ∂RE1/∂ũf = 1 in the response function box and is zero elsewhere, and ∂RE1/∂υ̃f = 0; for the meridional environmental wind response function, ∂RE2/∂υ̃f = 1 in the response function box and is zero elsewhere, and ∂RE2/∂ũf = 0.

Fig. 3.

(a) Flowchart describing how the response functions RE1 and RE2 are calculated from final time forecast vorticity ζf . Here E is the environmental projection operator. The operator that inverts vorticity and calculates the horizontal wind attributed to that vorticity is Q−1. The horizontal wind field attributed to vorticity outside of the response function box is ũf , υ̃f . (b) Flowchart describing how the initial conditions to the adjoint model, ∂RE1/∂ζf and ∂RE1/∂ζf , are calculated from ∂RE1/∂ũf and ∂RE1/∂f . Operators E* and (Q−1)* are the adjoints of E and Q−1, respectively. Note that for the zonal environmental wind response function ∂RE1/∂ũf = 1 in the response function box and is zero elsewhere, and ∂RE1/∂υ̃f = 0; for the meridional environmental wind response function, ∂RE2/∂υ̃f = 1 in the response function box and is zero elsewhere, and ∂RE2/∂ũf = 0.

Since all vorticity within the response function box is removed when the environmental wind is calculated, small perturbations to the final-time location of the BSV within the response function box have no effect on RE1 and RE2. A translation of the BSV caused by advection by the perturbation wind field is manifest as a dipole of positive and negative perturbation vorticity oriented in the direction of the translation (Fig. 1e). As long as the vorticity associated with the TC is not advected outside of the response function box, any translation of the TC within the box has no effect on the response functions. This assumption is less restrictive than the previous assumption that the TC had to remain in the center of the response function box.

b. Adjoint sensitivity gradients for RE1

By construction, the distribution of ∂RE1/∂ζ that is used to initialize the adjoint model (Fig. 1b) is identical to the initialized sensitivity of R1 (Fig. 1a), except that sensitivity of RE1 to vorticity is zero within the response function box. Since all vorticity within the response function box is removed in order to calculate ũ, perturbations to vorticity within the response function box can have no effect on RE1 at this time.

The structure of ∂RE1/∂ζ at time t = 0 (Fig. 1d) looks quite different from that of R1 (Fig. 1c). Maximum sensitivities still appear to the northeast and southwest of the BSV, but the sensitivities west and east of the BSV associated with the PCD effect have vanished. In fact, there is weak sensitivity of the opposite sign present west and east of the BSV, compared to the sensitivity of R1. The differences in the ∂R1/∂ζ and ∂RE1/∂ζ fields at adjoint model initialization (Figs. 1c and 1d, respectively) appear primarily directly west and east of the BSV (Fig. 4a) and are related to the PCD effect.

Fig. 4.

(a) Basic state vorticity contoured every 2.0 × 10−5 s−1 and difference between ∂RE1/∂ζ and ∂R1/∂ζ shaded every 5.0 × 104 m at model initialization. (b) Basic state vorticity (contoured) and perturbation vorticity shaded every 5.0 × 10−7 s−1 at model initialization. (c) Perturbation vorticity (shaded) and perturbation winds at model verification. (d) Perturbation environmental vorticity (shaded) and perturbation environmental winds at model verification. The box at the center of the plot corresponds to the boundaries of the response function box.

Fig. 4.

(a) Basic state vorticity contoured every 2.0 × 10−5 s−1 and difference between ∂RE1/∂ζ and ∂R1/∂ζ shaded every 5.0 × 104 m at model initialization. (b) Basic state vorticity (contoured) and perturbation vorticity shaded every 5.0 × 10−7 s−1 at model initialization. (c) Perturbation vorticity (shaded) and perturbation winds at model verification. (d) Perturbation environmental vorticity (shaded) and perturbation environmental winds at model verification. The box at the center of the plot corresponds to the boundaries of the response function box.

c. Perturbation test for validation of dynamical interpretation

A test similar to that performed for the dynamical interpretation of ∂R1/∂ζ (section 3b) is performed here for dynamical interpretation of ∂RE1/∂ζ. The sensitivity gradient of RE1 with respect to vorticity is scaled to create an initial condition perturbation vorticity field that will increase RE1. The nonlinear model is then integrated forward 24 h with the perturbed initial conditions.

The perturbation environmental vorticity and perturbation environmental wind at 24 h reveal that the perturbations to increase RE1 create a uniform zonal perturbation environmental flow in the response function box (Fig. 1f). After the perturbation vorticity within the response function box is removed, the vorticity that remains (constituting the environmental vorticity) appears as a positive (negative) gyre directly north (south) of the response function box. Inversion of this environmental vorticity yields a zonal perturbation environmental wind in the response function box. Thus, the zonal steering of the TC, defined as the hurricane advection flow due to vorticity outside of the response function box, has been increased.

A test of linearity similar to that performed for the previous perturbation case is performed. For this case, ΔRE1 = 0.416 m s−1 and the approximation of this value assuming linearity is δRE1 = 0.420 m s−1, indicating that the adjoint model accounts for 99.0% of the change in RE1.

d. Elimination of PCD effect in RE1

The reversal in sign between ∂R1/∂ζ and ∂RE1/∂ζ in these regions west and east of the BSV indicates that positive (negative) perturbation vorticity introduced west (east) of the BSV at model initialization will increase R1 and decrease RE1. To show that the PCD effect is eliminated from ∂RE1/∂ζ, another perturbation run is performed with a positive (negative) perturbation vortex introduced west (east) of the BSV at the initialization of the nonlinear forward model (Fig. 4b). These perturbation vortices have a maximum amplitude of 4.5 × 10−6 s−1. Perturbation vorticity and winds are calculated at the end of the 24-h model integration, and the response functions R1 (Fig. 4c) and RE1 (Fig. 4d) are evaluated.

Perturbation vorticity and winds contributing to R1 (Fig. 4c) show a definite PCD effect. The dipole in perturbation vorticity in the response function box is a clear indication that the BSV has been advected to the northwest by the perturbation wind field by this time. Likewise, the perturbation vortices introduced at model initialization have been advected cyclonically around the BSV by the basic state wind field, appearing to the southwest and northeast of the response function box. The average perturbation zonal wind in the response function box is positive, with ΔR1 = 0.101 m s−1, while δR1 = 0.101 m s−1, with the adjoint model accounting for 99.6% of the change in R1.

By removing any perturbation vorticity in the response function box, the perturbation environmental vorticity and perturbation environmental winds contributing to RE1 (Fig. 4d) reveal a reversal in the direction of zonal flow in the box. All significant perturbation vorticity associated with the PCD effect existed within the response function box. Since all perturbation vorticity in the response function box is removed to calculate RE1, the PCD effect has no influence on the calculation of RE1. The perturbation vorticity on the southwest and northeast corners of the response function box constitutes the environmental vorticity. This environmental vorticity induces an environmental zonal wind that is negative in the response function box, with ΔRE1 = −0.141 m s−1. The approximation of ΔRE1 assuming linearity is δRE1 = −0.138 m s−1, indicating that the adjoint model could account for 97.9% of the change in RE1.

This analysis validates the dynamical interpretation of ∂R1/∂ζ and ∂RE1/∂ζ fields, and how those differences relate to the PCD effect. Sensitivities of R1 west and east of the BSV are of opposite sign compared to sensitivities of RE1 because, for the perturbations considered, the PCD effect introduces positive perturbation flow in the response function box, while the vorticity that constitutes the “environment” introduces negative perturbation flow in the response function box. The PCD effect dwarfs the effect of the environmental vorticity, making ΔR1 positive. Since the PCD effect is eliminated in the calculation of RE1, only the influence of the environmental vorticity remains, making ΔRE1 negative.

This methodology comes with its own limitations. The response function box separates vorticity associated with the TC (vorticity inside the box) from vorticity associated with the environment (vorticity outside the box). Any vorticity of an environmental feature that migrates into the box will be zeroed out in the response function and not contribute to the hurricane advection flow assumed to steer the TC. Thus, any interaction of the TC with its environment that produces environmental vorticity inside of the response function box at model verification will not be manifest in the sensitivities.

5. Conclusions

It has been shown that the definition of TC steering as a deep layer mean wind averaged around a TC center is not suitable for defining response functions to calculate adjoint-derived sensitivities of TC steering. The problem is that the deep-layer mean wind averaged around the TC is a measure of the environmental flow that steers the TC only so long as the TC is in the center of the averaging box, such that the symmetric circulation about the TC is removed in the averaging. This definition has been validated when calculating the steering flow for both observed and modeled TCs (Chan 2005). However, when the calculation is used to define a response function for the purpose of calculating adjoint-derived forecast sensitivities of TC steering, the assumption that the TC remains in the same location is often violated. Since the response functions R1 and R2 can be influenced by perturbing either the environmental flow in the vicinity of the TC (a change to the steering) or the final time location of the TC within the response function box (not a change to the steering), these sensitivity gradients do not necessarily correspond to sensitivities of instantaneous TC steering.

A solution is proposed and validated in the context of a two-dimensional, nondivergent barotropic model by defining new response functions that eliminate the PCD effect. By defining the steering of the TC by the “hurricane advection flow”, which is an average of the environmental flow in the vicinity of the TC rather than an average of the full flow, the PCD effect is removed. While R1 and R2 require that the TC remain in the center of the response function box, the new response functions RE1 and RE2 only require that the vorticity associated with the TC remain within the response function box.

While the solution presented here has been validated within the context of a barotropic model, the RE1 and RE2 response functions should be tested in real TC simulations using a full-physics numerical weather prediction (NWP) model and that model’s adjoint. Clearly, issues regarding the validity of the assumption of linearity in such a model will include issues of the effect of diabatic processes that are not present in the simple model. While the relationship between the adjoint-derived sensitivity gradients and the physics of the tangent-linear model cannot be as clear in a full-physics model as they are in the simple model, it is expected that the dynamical interpretations of sensitivity gradients provided in this study will be a helpful corollary in a more complicated adjoint model. Real TCs are steered by a complexly evolving environment that can itself be sensitive to small changes to the initial conditions; in such a case one would expect the steering flow of the TC to be sensitive to the TC vortex as well as to relevant aspects of the environment. The results of this ongoing work will be provided in a future study.

A future application of this methodology includes the use of these adjoint-derived sensitivities to define targets for targeted observations with the explicit goal of improving TC track prediction. Dynamical sensitivities of TC steering to model initial conditions provide the requisite a priori information about where perturbations to the initial conditions of an NWP model will have the strongest effect on TC steering specifically. To apply this technique in an operational context, it would be necessary to determine an appropriate response function box size. The box needs to be large enough to accommodate the migration of the TC as a result of perturbations to the initial conditions, but small enough to produce meaningful results.

Dynamical sensitivities alone are insufficient to make well-informed choices about adaptive observation targeting. Dynamical sensitivity must be combined with information about the uncertainty in the initial conditions and information about how a given observation would be assimilated into the initial conditions. Coupling the output of an adjoint model with the adjoint of a data assimilation system would allow for the calculation of sensitivities of a response function to (individual) observations (Langland and Baker 2004). By combining a priori information about the dynamical sensitivity to the model state, the uncertainty in the initial conditions, and the impact of a given observation assimilated into the initial conditions, sensitivities of an appropriate response function for TC steering can be utilized in a truly objective targeting strategy to improve TC track forecasting.

Acknowledgments

This material is based on work supported by the National Science Foundation under Grant 0529343 and the Office of Naval Research under Grant N000141610563PR.

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Footnotes

Corresponding author address: Brett T. Hoover, Department of Atmospheric and Oceanic Sciences, University of Wisconsin—Madison, 1225 W. Dayton St., Madison, WI 53706. Email: bthoover@wisc.edu

1

Most often, sensitivities are calculated with respect to the model initial and boundary conditions; however, these dynamical sensitivities may be calculated with respect to the model forecast trajectory, xτ 0 < τ < tf , as well. For the purposes of this study, references to x can be inferred to mean xf when referring to the response function, while references to x can be inferred to mean xo when referring to sensitivities of the response function.

2

The code and supporting documentation are available from the first author upon request.

3

Note that ∂R1/∂ζ is computed at initialization of the adjoint model by defining ∂R1/∂u, ∂R1/∂υ directly and computing ∂R1/∂ζ by using the adjoint of the successive overrelaxation scheme used to compute u, υ from ζ.

4

Perturbation zonal flow attributed to the northward displacement of the BSV accounts for 55% of the perturbation zonal flow in the response function box.