We obtain TC best-track data from the National Hurricane Center (NHC) and the United States Navy Joint Typhoon Warning Center (JTWC). Best-track data are produced in post-season analysis using all available observations, e.g., satellite microwave imagery and aircraft reconnaissance, to generate the best estimate of TC location and intensity. Data are 6-hourly with occasional extra time points for landfall. The NHC Hurricane Database (HURDAT2) provides best-track data for the North Atlantic and eastern Pacific basins . The best-track data for the western Pacific basin, North Indian Ocean basin, and basins in the Southern Hemisphere are obtained from the JTWC . Together, these five regions capture all TC activity on Earth. We chose to use best-track data from the NHC and the JTWC because both agencies provide TC intensity estimates based upon a common definition of 1 min, 10 m maximum sustained wind, rounded to the nearest 5 kn. A final, important remark when considering data from the different ocean basins is the fact that aircraft reconnaissance is present in the North Atlantic but largely absent in the other basins, likely providing more accurate TC intensity estimates in the North Atlantic compared to elsewhere.

We consider all TCs from 2007 to 2018. The start year of 2007 is chosen to ensure sufficient coverage by Argo floats, detailed in the following section. Over this span of 12 years, we have a total of 1089 tracks, with 191 in the North Atlantic, 223 in the western Pacific, 339 in the eastern Pacific, 66 in the North Indian Ocean, and 270 in the Southern Hemisphere. These tracks are depicted in Fig. 3a.

Argo is a network of autonomous floats sampling global subsurface ocean temperature and salinity, with each float reporting roughly every 10 d. Argo floats have been deployed since the early 2000s and reached the desired nominal distribution of one float for every 3° longitude by 3° latitude box in the global ice-free ocean in 2007 , providing unprecedented subsurface spatial and temporal coverage, with no seasonal sampling bias . The geographical distribution of Argo floats is depicted in Fig. a. Each Argo observation consists of temperature and salinity measurements, indexed by pressure, taken as the Argo float ascends to the surface from a depth of roughly 2000 m. The resulting vertical profile is associated with a time stamp as well as satellite-determined latitude and longitude coordinates. Figure b plots two such temperature profiles, reported in roughly the same location before and after the passage of Hurricane Maria (2017). The leftward shift from the first profile to the second (denoted with an orange arrow) illustrates the reduction in thermal energy associated with Hurricane Maria's passage. The green arrow illustrates the deepening of the thermocline, the change point between an upper, well-mixed layer and cooler subsurface waters, as a result of increased mixing. For the rest of the paper, we will interchangeably use the terms “depth”, which is more familiar, and “pressure”, which is more accurate. Argo floats record pressure in decibars (dbar), and a 1 dbar change in pressure corresponds to approximately 1 m change in depth .

For our purposes, the chief contribution of Argo to our oceanographic and meteorological pursuits is its provision of vertical information, from the surface of the ocean to roughly 2000 m below the surface, in tandem with its global coverage, with one float per 3° longitude by 3° latitude box on average . The vertical information from the Argo profiles, unattainable by remote sensing methods such as satellites, is key to quantifying the thermal response of the ocean to the passage of a TC, which varies with depth. Argo profiles, however, are not without shortcomings for our purposes; their spatiotemporal resolution is coarser than typical TC spatial scales and timescales, which necessitates the statistical methods developed herein.

An Argo float takes a profile in the following fashion. Every sampling period, typically every 10 d, the float descends from its “parking” depth of 1000 m to a profiling depth of 2000 m and then ascends from 2000 m to the surface over the course of about 6 h, taking observations of temperature and salinity along the way . It then rests at the surface for 15 min to 1 h, determining its location and communicating its measurements via satellite. Finally, the float returns to its parking depth until its next sampling period to minimize float drift.

In this study, we refer to an individual Argo device as a “float” and to the measurements taken vertically at a certain longitude, latitude, and time as a “profile”. Each Argo profile is uniquely identified by its float identifier (unique to an Argo float) and cycle number (for a given float, unique to each profile).

To quantify the ocean thermal response to the passage of a TC, we perform a pairing procedure inspired by . Intuitively, we seek pairs of profiles which “straddle” the passage of TC, i.e., one observation before and one observation during or after the TC passage. These Argo profile pairs are placed in a TC-centric coordinate system through a projection process, which rewrites the space–time coordinates of an Argo profile pair in reference to a passing tropical cyclone. Figure  illustrates this process using the TC track and profile pairs associated with Hurricane Maria (2017), and the full set of profile pairs in the TC-centric coordinate system is depicted in Fig. b. The pairing and projection procedures are formalized in Sect. .

In this section, we describe two key preprocessing steps performed on the Argo profiles.

To associate Argo profiles with tropical cyclones, we perform a two-step procedure of pairing and projection. In the pairing step, we identify Argo profiles that are close to each other and straddle a passing tropical cyclone such that one profile is observed before the storm to characterize the pre-storm ocean state (referred to as the baseline profile), and another is observed during or after the storm to characterize the change in the ocean state (referred to as the signal profile). In the projection step, we change the coordinate system of the Argo profile pairs to one parameterized in terms of the passing tropical cyclone. This is done by performing an orthogonal projection of the Argo profile pair onto the tropical cyclone track. Both procedures are formalized in the sequel.

We first establish notation. Define the Earth coordinates as ((xlon,xlat),tsignal;tbaseline), i.e., the longitude, latitude, and time that the baseline and signal profiles were taken, and the TC-centric coordinates as (d,τ), where d is the signed great circle angular distance on Earth between (xlon,xlat) and π(xlon,xlat), the orthogonal projection onto the tropical cyclone track, and τ is the time difference between tsignal and tπ(xlon,xlat), the time when the tropical cyclone was at π(xlon,xlat). Due to the closeness constraints that we impose between the baseline and signal profiles, described in the following subsection, only one set of longitude–latitude coordinates is necessary for both profiles.

Although the temperature differences induced by tropical cyclone passage occur on relatively small temporal scales, there is a nontrivial seasonal effect experienced by the ocean during the tropical cyclone season, at the timescales considered in our analysis. We focus here on the methodology for estimating the seasonal cycle and fully detail the resulting empirical findings in Sect. . In order to properly account for the seasonal effect, we adapt previous methodology to estimate a global mean field specifically for the “counterfactual”; i.e., we characterize the state of ocean temperatures during the TC season, using the non-TC profiles of Sect. .

Using the non-TC profiles of Sect. , we estimate, separately for each pressure level z=10,20,…,200 and for the vertically averaged temperature values, a global mean field through a variant of the Roemmich–Gilson climatology. The Roemmich–Gilson climatology was among the first to take advantage of the spatial coverage of the Argo array to characterize the mean state and annual cycle of temperature in the global upper ocean. Specifically, they used a local linear regression model to estimate the climatological mean temperature as a function of location (xlon,xlat), pressure level z, and day of year t. This model is written down explicitly in , and our model, which differs marginally from that of in that we do not weight observations based upon the distance from the center point (xlon*,xlat*) or use data from pressure levels z′≠z to form estimates at pressure level z, is elucidated in Eq. ().

At each of these 20 pressure levels (or for the vertical average) and at each grid point of a mesh consisting of 180 equally spaced latitude points and 360 equally spaced longitude points, we fit 6m(xlon,xlat,t)=β0+[first- and second-order terms inxlatandxlon]+∑k=16γksin⁡2πkt365.25+∑k=16δkcos⁡2πkt365.25 to the temperature observations, using all the non-TC profiles that fall within ±8° (longitude–latitude) of the (xlon*,xlat*) point at which our local regression function is centered. Since these models are fit separately over each window and depth, the estimation procedure can be trivially parallelized.

Let us denote by mxlon*,xlat* the mean function derived by fitting a local linear regression with data within a window centered at (xlon*,xlat*) using Eq. () for a fixed depth z or for the vertically averaged temperature values. Then, for each of the TC profiles (Sect. ), we produce the seasonal mean-adjusted temperature value in the following fashion. For a profile located at x=(xlon,xlat) at time t with temperature value T, obtain the closest grid point at which a local linear function was fitted: 7(xlon′,xlat′)=arg⁡min⁡x′‖x′-x‖22. Then, at each pressure level z=10,20,…,200, we obtain the mean-adjusted profiles: 8T̃=T-mxlon′,xlat′(xlon′,xlat′,t). For each temperature value T in the TC profile set, we apply this mean adjustment to obtain the seasonally adjusted temperature value T̃. In particular, for a baseline–signal temperature pair Tbaseline,Tsignal, occurring at times tbaseline,tsignal, we obtain the seasonally adjusted temperature value pair T̃baseline,T̃signal. This gives rise to the seasonally adjusted temperature differences 9y=T̃signal-T̃baseline. In terms of the high-level model Eq. (), Eq. () in fact represents the temperature difference without the seasonal effect. This may be seen through the decomposition: y=T̃signal-T̃baseline=Tsignal-mxlon′,xlat′(xlon′,xlat′,tsignal)-(Tbaseline-mxlon′,xlat′(xlon′,xlat′,tbaseline))=(Tsignal-Tbaseline)-(mxlon′,xlat′(xlon′,xlat′,tsignal)-mxlon′,xlat′(xlon′,xlat′,tbaseline))=temperature difference-seasonal effect≜seasonally adjusted temperature difference, since the last two mean field terms only differ in terms of the contribution of the seasonal harmonics in Eq. ().

Having accounted for the non-stochastic seasonal term, we may now directly analyze the reduced model. This is informally described as 10seasonally adjusted temperature difference=TC signal+ocean variability and formally described, in terms of the ANOVA decomposition, as 11Tsignal-Tbaseline-(msignal-mbaseline)=s(d,τ)+(a(x,tsignal)-a(x,tbaseline)), where we have economized the notation from Eqs. () and () for brevity. The fitted seasonal means, at a depth of z=40, are depicted in Fig. , with the full set of estimates across all pressure levels deferred to the Supplement. The practical effect of this seasonal adjustment is illustrated in Sect. . The mean field adjustment is also performed for the non-TC profiles, yielding mean zero observations for the purposes of fitting the Gaussian process model of Sect. .

Having obtained seasonally adjusted temperature differences at all pressure levels, we next detail our method for estimating the variability of these temperature differences. Our general approach is an adaptation of the Gaussian process model proposed by , which handles the globally nonstationary covariance structure of the ocean using locally stationary Gaussian processes in a moving-window fashion . As in Sect. , we learn the parameters of our model using the non-TC profiles, in order to capture the local dynamics of the ocean without TC-induced effects. However, once fitted, the model is used to provide information about ocean variability in the time and locations of the TC profiles. This is a subtle, but important, distinction.

Our local Gaussian process models are fitted on an equally spaced grid of 180 latitude points and 360 longitude points, with a time window centered on September. This prioritizes fidelity towards ocean dynamics during the late summer and early fall months, when tropical cyclones are prevalent in the Northern Hemisphere, where the majority of the TC profile pairs are observed. As before, this analysis is carried out separately at each pressure level z=10,…,200 and for the vertically averaged temperature values.

In the following exposition, we follow the notation of , with slight adaptations to maintain consistency with Sect. . We use the term mean-adjusted interchangeably with seasonally adjusted to emphasize that the mean-adjusted temperature values are zero-mean (in the absence of a TC signal), as required by the Gaussian process model assumptions.

Let T̃ denote a mean-adjusted temperature value associated with a non-TC Argo profile at a fixed pressure level. At each grid point (xlon*,xlat*) and fixed pressure level z, we assume the following model for data points T̃ falling spatially within a window of ±5° of the grid point (xlon*,xlat*) and temporally within the 3 months centered on September: 12T̃i,j=fi(xi,j,ti,j)+εi,j,fi∼iidGP(0,k(x1,t1,x2,t2;θ)), where i=1,…,n indexes the years and j=1,…,mi indexes the observations within the geospatial window for each year, xi,j is the spatial location associated with the mean-adjusted temperature value T̃i,j, and ti,j is the time at which it was observed. The εi,j is a Gaussian nugget effect , with εi,j∼iidN(0,σ2), and GP(0,k(x1,t1,x2,t2;θ)) denotes a zero-mean Gaussian process with a stationary covariance function with parameters θ=(ϕ,θlat,θlon,θt). The specific function used is an anisotropic geospatial covariance function 13k(x1,t1,x2,t2;θ)=ϕexp⁡{-d(x1,t1,x2,t2)} with ϕ>0, where 14d(x1,t1,x2,t2)=xlat,1-xlat,2θlat2+xlon,1-xlon,2θlon2+t1-t2θt2, with θlat,θlon,θt>0. As in , these parameters are obtained through the method of maximum likelihood. As with the local mean field, the estimation of these parameters may be done in parallel across windows and pressure levels.

A few remarks are in order. First, we note that the fitted parameters ϕ(x), θlat(x), θlon(x), θt(x), σ(x) are all functions of x=(xlon,xlat), as we assume local stationarity to capture global nonstationarity. At each location, the Gaussian process model used to estimate these parameters is centered on September, as most TCs in our database occur during this time period. We then use these parameters to model the variance structure of profile pairs in all ocean basins; one could refine these estimates by fitting a model centered about every month and using the model whose central month is closest in time to each profile pair. Finally, we observe that, by construction, the profile pairs from Sect.  are assumed to have the same locations, i.e., xlat,1=xlat,2 and xlon,1=xlon,2. Therefore, the covariance between the two profiles in a profile pair is given by the following reduced expressions. For a mean-adjusted temperature value pair T̃baseline,T̃signal located at (xlon,xlat) with times tbaseline,tsignal, let x′=(xlon′,xlat′) be the nearest grid point, as in Sect. , and let ϕ,θt,σ as in Eq. () be 15var(T̃baseline)=var(T̃signal)=ϕ(x′)+(σ(x′))2;16cov(T̃baseline,T̃signal)=ϕ(x′)exp⁡-tbaseline-tsignalθt(x′).

To give an impression of the general variability in the global ocean, we plot the fitted ϕ parameters, for a fixed pressure level z=40, in Fig. . The parameters ϕ are plotted because ϕ is the dominant term in each variability estimate. The full set of figures depicting the fitted parameters and temperature difference variances, at each pressure level, is deferred to Sect. S5 of the Supplement.

We note that, even though only ϕ, θt, and σ are used to evaluate the data covariances, the full model (Eqs. , , and ) must be fitted to estimate these parameters, and we require the full Argo database, including the non-TC profiles, to obtain these estimates.

For simplicity, we assume that two profiles which do not share the same baseline profile are uncorrelated. However, there are cases in which the lineage of a baseline profile includes multiple signal profiles. In these cases, we account for the covariance between the profiles in the lineage. We detail the case in which there are two signal profiles with mean-adjusted temperature values T̃signal(j),T̃signal(k), associated with a baseline profile with a mean-adjusted temperature value T̃baseline(i), but the following framework is trivially extended to an arbitrary number of signal profiles in the lineage. In practice, there will be no more than six signal profiles for a given baseline profile, due to the separation constraint introduced in Sect. . Define the shorthand T̃(ijk)=T̃baseline(i)T̃signal(j)T̃signal(k)⊤. We suppress the location x′ at which the parameters ϕ,θt,σ are evaluated, for brevity. First, consider the covariance structure between the mean-adjusted temperature values of the three profiles in this lineage (one baseline T̃baseline(i) and two signal T̃signal(j),T̃signal(k) values, occurring at times ti,tj,tk): 17cov(T̃(ijk))=ϕ+σ2ϕexp⁡{-|tj-tiθt|}ϕexp⁡{-|tk-tiθt|}ϕexp⁡{-|ti-tjθt|}ϕ+σ2ϕexp⁡{-|tk-tjθt|}ϕexp⁡{-|ti-tkθt|}ϕexp⁡{-|tj-tkθt|}ϕ+σ2. The covariance matrix for the temperature differences may then be obtained as follows. Define the mean-adjusted temperature differences 18y(ij)=T̃signal(j)-T̃baseline(i)19y(ik)=T̃signal(k)-T̃baseline(i) and the shorthand y(ijk)=y(ij)y(ik)⊤. Observe that y(ijk) is simply a linear transformation of T̃(ijk): 20y(ijk)=y(ij)y(ik)=-110-101T̃baseline(i)T̃signal(j)T̃signal(k). In general, the transformation matrix is a column of minus ones horizontally concatenated with the identity matrix. It immediately follows that 21cov(y(ijk))=-110-101cov(T̃(ijk))-110-101⊤. Recall that, because the temperature differences have been mean-adjusted, they are assumed to be zero-mean in the absence of a TC signal. Therefore, when there is no TC signal, the distribution of y(ijk) is Gaussian with mean zero and covariance cov(y(ijk)), given in Eq. (). Mathematically speaking, we have accounted for the (a(x,tsignal)-a(x,tbaseline)) variability term of Eq. (). This forms the basis for the estimation of s(d,τ), our key quantity of interest, in Sect. .

Having learned a model for the (a(x,tsignal)-a(x,tbaseline)) term of Eq. () through the Gaussian process model of Sect. , we find ourselves properly positioned to estimate the TC-induced signal s(d,τ). This we do with a thin-plate spline, a flexible nonparametric technique for fitting a surface in two dimensions, subject to a curvature penalty and observation weights. Specifically, we smooth the seasonally adjusted temperature differences using a fixed-knot thin-plate spline smoother , in which we use the covariances estimated in Sect.  to account for heteroskedasticity and cross-observation dependence . As with the local mean field model of Sect.  and the Gaussian process model of Sect. , this procedure is conducted separately at each pressure level z∈{10,20,…,200} and for the vertically averaged profiles.

We use a fixed-knot smoother (rather than placing a knot at every design point), chiefly in order to improve the conditioning of the linear operators. As an additional, computational consideration, the use of fixed knots reduces the time and space complexity of the procedure. The fixed knots are equally spaced in the [-8,+8]×[-2,+20] domain, in a grid formed from the cross-product of 33 points on the cross-track axis and 45 points on the time axis. This yields a knot every 0.5 units along each dimension, including on the boundary.

The weighting of observations by the covariance between temperature differences is also noteworthy. From a data model standpoint, it is known that we are using observations with drastically different noise levels a(xlon,xlat,tbaseline), a(xlon,xlat,tsignal), due to the effect of time and location. Since we have estimated these noise levels in Sect. , the covariance reweighting allows us to directly incorporate this information when estimating s(d,τ) into the reduced model Eq. (). From a practical standpoint, the covariance reweighting has a real effect in improving the signal-to-noise ratio, as shown in Fig.  of Sect. . The full details of the thin-plate spline smoother, including its motivation and derivation, are given in Appendix .

Let ξ1,…,ξn∈Ω⊂R2 be the design points and y1,…,yn∈R be the corresponding responses. We further fix a grid of regularly spaced knots, ξ̃1,…,ξ̃m∈Ω, m≪n, on which the basis functions of our smoother will be centered. We use the shorthand notation Ξ=ξ1…ξn⊤, Ξ̃=ξ̃1…ξ̃n⊤, and y=y1…yn⊤. Finally, we denote by W∈Rn×n a weight matrix. Traditionally, this is the inverse covariance matrix of the observations y.

In our usage, we take ξ1,…,ξn to be the (d,τ) coordinates of our Argo profile pairs, as obtained from Sect. ; y1,…,yn to be the corresponding seasonally adjusted temperature differences y of those pairs, as obtained from Sect. ; and W=cov(y)-1 to be the inverse matrix of covariances between those seasonally adjusted temperature differences, as obtained from Sect. . Therefore, here Ω=[-8,+8]×[-2,+20], where the first dimension (cross-track angle) is measured in degrees and the second dimension (time since TC passage) is measured in days.

The fixed-knot covariance-reweighted thin-plate spline problem is posed as follows. We seek coefficients δ,β such that the following objective is minimized: 22(y-R̃δ-L⊤β)⊤W(y-R̃δ-L⊤β)+λδ⊤Sδ. The first term in Eq. () is a generalized least-squares data-fit criterion, and the second term is a roughness penalty , with R̃,L,S defined in the sequel. The parameter λ>0 mediates the competing demands of the data fidelity and roughness penalty terms.

First, we detail the effect of the seasonal mean field adjustment, described in Sect. , on the observed data. Recall that the model () is fitted separately at each pressure level z=10,…,200. In Fig. , we depict the effect of the seasonal adjustment on temperature differences at a pressure level of z=40 dbar. The raw temperature differences are displayed in Fig. a, with the seasonal differences plotted in Fig. b and the seasonally adjusted temperature differences in Fig. c. The latter is the result of subtracting (b) from (a). The individual differences are subject to an isotropic Gaussian kernel smoother with a bandwidth of σ=0.2, in order to more clearly display the data.

Figure b shows a warming effect that increases in magnitude as the time between the signal profile and TC passage increases. This is consistent with the notion that the ocean experiences an overall warming effect at this depth during the TC season. Note that because we are plotting the difference in the mean field evaluated at the times and locations of actual TC-associated profile pairs, we are only examining the mean field differences during the tropical cyclone season of each ocean basin analyzed. Similar to Fig. , here the fitted seasonal effects are pooled over the entire TC season (and from different locations on Earth), which makes the fitted differences appear “noisy” in the TC-centric coordinate system, but the fitted mean fields are smooth as a function of ambient space and time, as expected based upon Eq. ().

The warming effect in the seasonal mean field difference is a consistent feature in the pressure levels z=30 through z=150. The pressure levels z∈{10,20} do not show a warming pattern and might even exhibit a slight cooling effect, while the pressure levels z≥160 show only a small contribution of the seasonal signal to differences in paired profiles. These phenomena are consistent with the spatial structure of the seasonal cycle . An adjustment for seasonal warming was not performed in a previous analysis of the ocean thermal response using paired Argo profiles ; as a result, some of the warming effects previously observed over longer time spans may in fact have been due to seasonal variation. The full sequence of results at all 20 pressure levels is deferred to Sect. S1 of the Supplement.

Having adjusted the temperature differences for the seasonal change using the model detailed in Sect.  and accounted for the variability across profile pairs via Sect. , we now apply the thin-plate spline smoother to the seasonally adjusted temperature differences to obtain our main scientific result: a characterization of the ocean thermal response to the passage of hurricane-strength tropical cyclones as a continuous function of time and cross-track distance. This function is depicted for data from profile pairs associated with a hurricane-strength tropical cyclone (formally defined as having a sustained wind speed of at least 64 kn at tπ(xlon,xlat), the time of TC passage) at a pressure level of z=10 dbar in Fig. .

The thin-plate splines are fitted separately at each depth z. Within each z∈{10,20,…,200}, we select a tuning parameter λ^zcv guided by leave-one-out cross-validation (LOOCV). Importantly, we do not take the λ^min that directly minimizes the LOOCV score. Instead, we take the λ<λ^min that produces a LOOCV error slightly (1 %) larger than the minimum LOOCV error. This is done because heterogeneity in the thermal response signal causes a direct minimization of the LOOCV score to oversmooth the signal. The full details of the cross-validation procedure, including its mathematical statement, plots illustrating the estimated test error achieved by different λ, and the chosen λ^zcv, are deferred to Appendix .

The central Fig. b depicts the thin-plate spline fit, accounting for correlation between observations at a depth of z=10 dbar. Formally, the correlation adjustment is performed as follows. We obtain an estimate for Σ=cov(y) through the Gaussian process model of Sect.  and provide this to our covariance-reweighted thin-plate spline smoother of Sect.  by setting W=Σ-1. Recall that, by construction, Σ is a block diagonal matrix, with blocks no larger than six entries wide; therefore, Σ-1 may be efficiently computed. For a very small number of temperature differences, the fitted variance is extremely small, on the order of ≤exp⁡{-4.5}. We truncate the left tail and filter out these observations to prevent distortion of the fit. This omits only 488 observation pairs out of 16 025.

In Fig. c, we perform a pointwise level α=0.05 hypothesis test on the thin-plate spline fit, using the variance estimates derived in Sect. . The issues of performing a pointwise hypothesis test with N=40000 points are manifold, and we defer discussion of extensions through which we could make the significance test more sophisticated to Sect. . Nonetheless, we see in Fig. c a simple fact: that the estimated cooling effect induced by tropical cyclones is of a greater magnitude than the variability of the estimates. Moreover, this cooling effect persists for a sustained period of time, stretching into 2 weeks following the tropical cyclone passage. We also see a rightward bias in the cooling effect, which may be attributed to the storm's direction of rotation.

Due to the Coriolis effect, TCs in the Northern Hemisphere will induce a greater cooling effect on the right side of the cross-track angle parameterization than on the left side, due to the combined effects of forward movement and counterclockwise wind rotation. TCs in the Southern Hemisphere, which experience clockwise wind rotation, will induce a greater cooling effect on the left side. In order to display this effect in a consistent manner across the hemispheres, we flip the sign of the cross-track angle for TCs in the Southern Hemisphere when producing Figs. –.

To illustrate the importance of estimating the correlations between observations for the final thin-plate spline fit, we plot in Fig. a the surface estimated by applying a thin-plate spline smoother without accounting for the variability estimates from Sect. , i.e., setting W to be the identity matrix. These estimates are tuned with the same LOOCV procedure. We observe a sustained warm anomaly at d≈-6 that is present over several days. Concerningly, this warm anomaly is comparable in magnitude with the expected cooling effect observed at d≈0. We also see negative anomalies which occur late in the recovery stage, in τ∈(18,20). When viewed in context with Fig. b and c, we see that adjusting for the correlations reduces the impact of some outlying observations causing these anomalous effects in the naive estimate, and the pointwise hypothesis test masks away the effects almost entirely.

An additional phenomenon of scientific interest that may be observed in the thin-plate spline fits is that of vertical mixing, in which the warm water close to the ocean surface is forced downwards by the force of the tropical cyclone . Although the net change in heat content incidental to the passage of a tropical cyclone is negative, vertical mixing may register as a warming effect at shallower subsurface ocean depths. This phenomenon may be observed in Fig. . In Fig. a, we recall the smoothed temperature differences at depth z=10, in which a dramatic cooling effect is depicted. At the deeper pressure level of z=60, however, we see a mixture of both cooling and warming; i.e., a net increase in temperature is observed at d∈(2,4), τ∈(0,6). Finally, we see in Fig. c that, by z=150, the warming effect has disappeared, while the cooling effect remains. We note that the characterization of this warming effect as a continuous function of both cross-track angle and time since TC passage is made possible by our methodological framework and was not possible in the previous literature .

The discussion of vertical mixing is further developed in Sect. , , and . The full set of covariance-reweighted thin-plate spline fits, at all pressure levels z=10,…,200, is deferred to Sect. S2 of the Supplement.

Of additional scientific interest is the characterization of the ocean thermal response for a fixed cross-track angle, as a function of time and pressure. To facilitate this analysis, we collect the 20 thin-plate spline fits for pressure levels z=10,…,200 from Sect.  and “marginalize” them along three sets of cross-track angles: d∈[-2.5,-1.5], d∈[-0.5,+0.5], and d∈[+1.5,+2.5]. These three plots are presented in Fig. . In each of the marginalized plots, we subset the covariance-reweighted thin-plate spline fits from Sect.  to each of the cross-track bins and average out the cross-track component. Variance estimates follow immediately (owing to the linearity of averaging) and are used to construct a pointwise α=0.05 hypothesis test. We display these marginalized estimates in Fig. .

The cross-track angle bins, centered on d=-2,0,+2, were chosen to emphasize three particular oceanographic and meteorological phenomena. First, the bin centered on d=0 in Fig. b depicts the ocean cooling induced by the passage of tropical cyclones. Of particular note is the fact that this cooling appears to persist for a sustained period of time, specifically up to at least 2 weeks. As one may expect, the cooling effect diminishes as pressure z or time since TC passage τ increases. Second, the bin centered at d=+2 in Fig. c depicts the phenomenon of vertical mixing previously discussed in Sect. . As TC winds mix surface water downward, we observe warming at depths between z=50 and z=150. Past z=150, we find that the warming effect has faded away. Finally, we contrast the strong warming effect observed in Fig. c with the lack of warming (at a statistically significant level) in Fig. a, which depicts the bin centered on d=-2. This reflects the fact that the effect of vertical mixing is asymmetric, due to the counterclockwise rotation of tropical cyclones in the Northern Hemisphere.

We further demonstrate that our continuous-time methodology captures the temporally binned results of , whose profile pairing process we used in this paper. To introduce temporal binning, we take the covariance-reweighted thin-plate spline estimates from Sect.  across pressure levels z=10,…,200 and split them into two bins, τ∈[0,+3) and τ∈[+3,+20), termed the “forced” stage and “recovery” stage by . For each of the two bins, we perform the same averaging process as in Sect.  to marginalize the estimates over the τ axis and obtain α=0.05 pointwise significance tests. We present the resulting fits as functions of cross-track angle d and pressure level z in Fig. .

Several noteworthy phenomena are present in Fig. . In Fig. a, we observe both the primary cooling effect and the secondary warming effect in the subsurface on the right side of the storm due to vertical mixing, as discussed in Sect.  and . We also observe a sustained cooling effect in Fig. b during the period called the “recovery” stage in . While the primary cooling effect observed in the two time periods is qualitatively consistent with the results presented by , their results also show a strong warming effect on both sides of the storm during the recovery stage (Fig. 10 in ), which is absent from our estimate. Overall, it appears that the recovery stage estimate in is “warm biased” in comparison to our estimate. We attribute the difference to our methodology's accounting of the seasonal mean field and variability estimates, which, when used in the thin-plate spline estimate, attenuate the influence of outlying observations dominated by non-TC oceanographic variability as illustrated in Fig. .

To provide a holistic understanding of the ocean thermal response structure as a function of depth, cross-track distance, and time since TC passage, we compute isosurface plots of the thin-plate spline estimates from Sect. . Two sets of isosurfaces are computed: one set for regions of negative temperature differences and one set for regions of positive temperature differences. These are shaded blue and red, respectively, in Fig. .

For each set of isosurfaces, we first mask the thin-plate spline estimates that fail to reject a pointwise level α=0.05 hypothesis test, using the variance estimates from Sect. . We then apply the marching cubes algorithm to identify positive and negative isosurfaces of magnitude 0.3 °C. The positive isosurface is colored in red and the negative isosurface in blue.

We observe in Fig.  a large blue mass corresponding to the TC-related ocean cooling. This blue mass is restricted to a relatively small range of cross-track angles near the center but persists in time. This behavior is consistent with Fig. . We also observe a smaller red mass, corresponding to the local warming effect associated with the vertical mixing induced by the TC, as previously seen in Fig.  and discussed in Sect. . Consistent with Figs. , , and , this warming effect is only observed on the positive side of the cross-track distance axis. The right panel of Fig.  mirrors Fig. a almost exactly, the only differences being the viewing angle and filtering of very small values inherent in finding nonzero isosurfaces.

Here, we present the results from the vertically averaged analysis, in which we estimate the full model Eqs. () and () using the vertically averaged temperature values from Sect. . As before, we choose the smoothing parameter λ^cv guided by leave-one-out cross-validation, with the full details deferred to Appendix .

The seasonally adjusted temperature differences and accompanying thin-plate spline fits are presented in Fig. . In Fig. a, we only show data from pairs associated with hurricane-strength TCs, in contrast to Fig. , which shows data from all profile pairs regardless of storm intensity. The restriction to hurricane-strength profile pairs here is to provide a fairer comparison with the thin-plate spline estimates of Fig. b and c, which use the hurricane-strength subset (as in Fig. ).

The thin-plate spline fit in Fig. b conforms to our intuition; we observe a cooling signal, as in Fig. b, but its magnitude is dampened by the vertical averaging procedure. More importantly, Fig. c does not show any warming effect indicative of vertical mixing. This is in contrast to Figs. b, c, and a, which treat depth as a separate dimension of analysis. From an oceanographic and meteorological perspective, this suggests that the vertical mixing-induced warming effect is local in depth; it only appears at some pressure levels, because it is caused by the forcing of warm surface waters downward by the TC. Once we integrate vertically, the local warming effect is cancelled out by a cooling effect near the surface and is no longer visible. From a statistical perspective, this underscores the importance of treating depth as a separate axis of analysis, using the vertical resolution of Argo observations.