Volume 45, Issue 18 p. 9739-9747
Research Letter
Free Access

Depth-Dependent Thermal Stress Around Corals in the Tropical Pacific Ocean

T. A. Schramek

Corresponding Author

T. A. Schramek

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USA

Correspondence to: T. A. Schramek,

[email protected]

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P. L. Colin

P. L. Colin

Coral Reef Research Foundation, Koror, Palau

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M. A. Merrifield

M. A. Merrifield

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USA

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E. J. Terrill

E. J. Terrill

Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, USA

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First published: 27 August 2018
Citations: 27

Abstract

Thermally driven bleaching events are a growing concern for reef ecosystems across the tropics. To assess and predict thermal stress impacts on reefs, remotely observed sea surface temperature (SST) commonly is used; however, reef communities typically extend to depths where SST alone may not be an accurate measure of in situ variability. Here nearly two decades of temperature observations (2- to 90-m depth) at three stations around Palau are used to develop an empirical model of temperature variability versus depth based on SST and sea level anomaly (SLA). The technique yields depth-averaged R2 values >0.88, with SLA predicting fore reef temperatures near the thermocline and SST capturing upper mixed layer temperatures. SLA complements SST by providing a proxy for vertical isotherm displacements driven by local and remote winds on intraseasonal to interannual time scales. Utilizing this concept, thermal stress on corals can be predicted from the surface through the mesophotic zone.

Key Points

  • Fore reef temperature in the tropics can be reconstructed with sea level and sea surface temperatures in multiple linear regressions
  • Sea surface temperature sets the temperature of the upper mixed layer, while sea level indicates temperatures near the thermocline
  • Thermal stress metrics can be adapted for mesophotic coral ecosystems by accounting for the variability in temperature near the thermocline

Plain Language Summary

Coral reefs are often bleached, leading to their death, due to exceedingly warm ocean temperatures. The temperature of the ocean's surface, measured globally by satellites, is often used as an indicator of the temperature and stress that corals experience, but it can only tell us what is happening near the surface. We present nearly two decades of temperature records from the reefs of Palau, an island nation in the tropical Pacific. This array of instruments was maintained by skilled divers routinely going deeper than 90 m. The observations allow us to show that the height of the ocean surface is a strong indicator of how ocean temperatures are changing tens of meters below. This can be coupled with observed sea surface temperature to predict the temperatures experienced by coral reefs living near the surface as well as those living deeper, down through the mesophotic zone, an area between 30 and 150 m deep. The research suggests that significant improvements can be made to how temperature stress on corals is assessed. We also find that thermal stress events can penetrate into the realm of deep mesophotic coral reefs, meaning that this zone might not be refugia for corals living in a warming ocean.

1 Introduction

Thermal stress is a major driver of coral bleaching, which threatens coral reefs around the world (Hughes et al., 2018; Langlais et al., 2017). Climate change and warming oceans adversely impact coral communities (Hughes et al., 2018; Wellington et al., 2001), and with no indication of immediate relief (Donner, 2009; Hoegh-Guldberg, 1999; Langlais et al., 2017), marine ecosystems (Glynn, 1993) and the social and economic systems dependent on them (Bridge et al., 2013) are threatened. Sea surface temperature (SST) is the principal ocean parameter used to predict coral bleaching (Gleeson & Strong, 1995; Liu et al., 2014, 2006; Strong et al., 2004) and to derive metrics quantifying thermal stress in surface waters, such as Degree Heating Weeks (DHW; Strong et al., 2004). The depth/temporal variability of thermal regimes on reef systems has been understudied leading to a lack of understanding of thermal stress on corals through the extent of their depth range. Deeper reef waters in the mesophotic zone (Baker et al., 2016; Kahng et al., 2010; Lesser et al., 2009), between 30 and 150 m, have been hypothesized to possibly provide refuge for coral reefs in the future (Bongaerts et al., 2017; Bridge et al., 2014; Neal et al., 2014; Riegl & Piller, 2003; Shlesinger et al., 2018) but have recently been shown to be ecologically distinct (Rocha et al., 2018). These factors motivate the need for new methods to assess coral stressors at each depth zone independently.

Warm conditions that may lead to thermal stress and coral bleaching currently are forecast using SST values derived from satellite observations (Heron et al., 2016; Liu et al., 2014, 2006). Elevated SSTs have been shown to be predictive of coral bleaching for shallow fore reef corals (Couch et al., 2017; Eakin et al., 2010; Glynn & D'croz, 1990; Liu et al., 2014; McClanahan et al., 2007). At the same time, quantifying accurate bleaching thresholds for corals between 10 and 30 m has been limited and assessments of thermal stress on mesophotic coral reefs have only been done in a few studies (Bruno et al., 2001; Nir et al., 2014). This is primarily due to the lack of sustained temperature records at depths >10–30 m and inconsistent surveying of bleaching events at depths exceeding 20 m (Bak et al., 2005). Depth has been recently shown to be a relevant parameter is assessing coral bleaching (Couch et al., 2017; Safaie et al., 2018) but in the context of other environmental and ecological factors. We propose that the vertical dimension, something not captured in SST measurements, is critical when developing indicators of thermal stress as temperature variance is observed to increase closer to the thermocline (Bak et al., 2005; Wolanski et al., 2004). Sea level anomaly (SLA) is a relevant observation that facilitates the prediction of temperatures starting at depths of 20–30 m, typically below the upper mixed layer (UML), for which SST is a poor proxy. A major part of the coral biomass exists below the base of the UML, denoted as the mixed layer depth (MLD), in the tropical Pacific (Bridge et al., 2013). The depth of isotherms below the MLD and SLA has been shown to be related in the tropical Pacific (Chaen & Wyrtki, 1981; Rebert et al., 1985), but the impact of isotherm displacements on the thermal structure at a tropical fore reef system has not been documented on the time scales considered here due to a lack of in situ observations.

The Republic of Palau, an island nation in the tropical western Pacific (Figure S1a in the supporting information), has a high diversity of marine habitats with exceptional coral reefs (Colin, 2009). Coral bleaching events have occurred in Palau with the largest on record concurrent with the 1998 La Niña (Bruno et al., 2001; Colin, 2009; Golbuu et al., 2007) with a lesser bleaching event on Palau's lagoons/outer slopes in 2010 (van Woesik et al., 2012). Bleaching in the mesophotic zone during the 1998 event was qualitatively observed down to 60 m (Bruno et al., 2001; Colin, 2009). Baker et al. (2016) documented bleaching during the 2010 El Niño–Southern Oscillation (ENSO) event in Palau. Qualitative examples of both shallow, depths <30 m, and mesophotic, depths 30–150 m, corals bleached during the 2010 event are shown in Figures 1c and 1d, respectively. During July 2014 there was some bleaching in shallow, depths 2–10 m, lagoon corals (Colin, personal observation). The summer of 2016 saw the initiation of minor bleaching on both the fore reefs and in the lagoons (Colin, personal observation). Impacted coral species during the 2014 and 2016 events are listed in the supporting information (Text S1). High bleaching occurrence at all depths where reefs occur during the 1998 event motivated a long-term temperature monitoring program by the Coral Reef Research Foundation using vertical arrays of thermographs. Program data from Short Drop Off, Ulong Rock, and West Channel (Table S1 and Figures S1c–S1e, respectively) provide a near two-decade record of the vertical temperature structure affecting Palau's fore reefs.

Details are in the caption following the image
(a) Mean temperature profiles from Palau (observed—solid lines, MLR modeled—dashed lines), at Short Drop Off (Figure S1c) on the East coast of Palau, during all El Niño (ONI > = 0.5—red) and La Niña (ONI < = −0.5—blue) events as well as a neutral condition (0.5 > ONI > −0.5—black). (b) Weekly SLA binned by ONI phase shows distinct regimes for the different phases of the ONI. Examples of coral bleached after the 2010 event from the (c) reef top (Acropora sp.) and (d) deeper reef slope (Pachyseris speciosa). (e) Linear regressions between SLA and the weekly averaged temperature observations at 57 m from SDO (Figure S1c), ULO (Figure S1d), and WCH (Figure S1e) with their corresponding R2 values noted. (g) Weekly averaged temperature observations from the same three locations as in (e) are shown. (h) MLR reconstructions (dashed lines) of each temperature record shown in (g) using tide gauge SLA and SST shown in Figures 3b and 3c. MLR = multiple linear regression; ONI = Oceanic Niño Index; SLA = sea level anomaly; SDO = Short Drop Off; ULO = Ulong Rock; WCH = West Channel.

There are two main objectives of this study: first, to assess the ability of SLA, coupled with SST and observed fore reef temperature, to reconstruct fore-reef temperature records and second, to apply DHW at a range of depths in the upper water column and adjust the thermal stress metric to account for increased temperature variance in depth.

2 Data and Methods

2.1 In Situ Observations and Gridded Products

In situ temperature at the seafloor was collected by the Coral Reef Research Foundation at 17 stations in three locations around Palau at depths ranging from 2 to 90 m (Table S1). Sampling intervals ranged from 30 s to 30 min. The observational campaign began in 2000 and is ongoing at the time of publication. Both daytime and nighttime records were used. A detailed description of the in situ data is available in the supporting information (Text S2 and Table S1).

SST at Palau was specified using the National Oceanic and Atmospheric Administration (NOAA) Coral Reef Watch (CRW) CoralTemp 5-km product (https://coralreefwatch.noaa.gov/satellite/coraltemp.php). SST time series were selected at the grid point closest to each observation station locations (Table S1; Heron et al., 2015, 2014; Liu et al., 2017). The SST product extends from 1985 to present. This product uses nighttime retrievals only from 2002 onward and is available as a part of CRW 5-km product that currently goes from 2013 to present for the Palau Virtual Station (https://coralreefwatch.noaa.gov/vs/data/palau.txt). The relevant climatological values used in this work were obtained from this Virtual Station output. Regional SST maps were constructed from the NOAA Optimum Interpolation SST V2 High Resolution Data Set due to ease of data access for the entire Pacific (https://www.esrl.noaa.gov/psd/data/gridded/data.noaa.oisst.v2.highres.html; Reynolds et al., 2007). Daily averages were obtained on a 0.25° × 0.25° global grid.

Daily tide gauge records were obtained from the Malakal Harbor station for 1970 through 2017 from the University of Hawaii Sea Level Center (http://uhslc.soest.hawaii.edu/data/; Caldwell et al., 2017). Regional sea level was specified using the global ocean gridded SLA L4 output, 0.25° × 0.25° resolution, from Copernicus Marine Environment Monitoring Service (marine.copernicus.eu/services-portfolio/access-to-products/). The National Centers for Environmental Prediction surface wind records (Kalnay et al., 1996) on a 0.25° × 0.25° global grid from 2000 to 2017 were obtained (https://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis.html).

2.2 Filtering and the Oceanic Niño Index

Weekly averages were constructed for all data records to examine subinertial variability. The local inertial period is roughly 4 days. Partitioning of the data into ENSO phases was done by finding concurrent time steps when the Oceanic Niño Index was greater than or equal to 0.5 (El Niño) or less than or equal to −0.5 (La Niña).

2.3 Reconstructing Temperature Using Regression Techniques

A multiple linear regression (MLR) analysis (Chatterjee & Hadi, 1986) was used to estimate in situ temperature based on SST (CoralTemp SST) and SLA (Malakal tide gauge). SLA2 was included as an MLR input to account for a nonlinear relationship between SLA and observed temperatures at some depth ranges (Figure 1e). The regression model, applied at each sensor depth, z,
urn:x-wiley:00948276:media:grl57896:grl57896-math-0001(1)
was used to reconstruct the upper ocean temperature profiles between 1985 and 2017. Code for implementing this methodology is available in the SI and can be accessed with accompanying example files at https://github.com/tschramek/ThermalStress.

2.4 Determining Thermal Stress-DHW

DHW is a measure of the sustained thermal stress experienced by coral communities (Liu et al., 2014; Strong et al., 2004). DHW is the sum of thermal anomalies (HotSpots; Heron et al., 2015) that exceed a prescribed threshold accumulated during a 12-week running window (Strong et al., 2004). Coral bleaching is known to occur when SSTs exceed 1°C above the maximum monthly mean (MMM) temperature (Glynn & D'croz, 1990); hence, MMM + 1°C is the typical threshold used for the DHW calculation or the bleaching threshold (Berkelmans & Willis, 1999).

To evaluate the potential thermal stress at depth, we considered three measures of DHW. The first is the traditional method based on when SST exceeds the bleaching threshold, which is computed using MMM at the sea surface, at z = 0, determined from SST:
urn:x-wiley:00948276:media:grl57896:grl57896-math-0002(2)
The second uses in situ temperature at each depth from the MLR reconstructions, with the bleaching threshold determined from SST,
urn:x-wiley:00948276:media:grl57896:grl57896-math-0003(3)
This method assumes that thermal stress tolerances appropriate for the sea surface apply at all depths. Lastly, we adjust the bleaching threshold with depth to account for changes in the mean and variable temperature through the water column. A scaled version of the bleaching threshold was computed as follows:
urn:x-wiley:00948276:media:grl57896:grl57896-math-0004(4)
Code for computing DHW has been provided in the SI and can be accessed with example data at https://github.com/tschramek/ThermalStress. MMM (z) were computed using the temperature reconstructions over the same base period (1985–2012) as the CRW supplied MMM computed from CoralTemp SSTs, noted above as MMM(0), for Palau (https://coralreefwatch.noaa.gov/vs/data/palau.txt).

3 Results

Changes in temperature structure with SLA are illustrated for different phases of ENSO, when SLAs range from high (La Niña) to low (El Niño) in Palau (Figures 1a and 1b). During La Niña conditions, the mixed layer is deep (greater than 35 m) with a weaker temperature gradient in the thermocline than during neutral ENSO phases (Baker et al., 2016). During El Niños, the mixed layer is shallow (15 m) with weak temperature gradients to ~35-m depth and strong gradients below ~35 m. In general, as sea level rises/falls, the thermocline deepens/shoals, with UML temperatures tracking SST.

Binning the temperature data by SST (Figure S2a) and SLA (Figure S2b) shows how the vertical temperature gradient varies with each variable. SST represents temperature in the UML; SLA captures temperature variability below the MLD. Binning of temperature observations by SST yields distinct temperature profiles with constant temperatures in depth to the MLD (Figure S2a). At three sites, small differences exist between the in situ temperature at 2-m depth and the CRW CoralTemp 5-km SST product (Figures S3a–S3c). Influences from the lagoon systems, be it rainfall runoff, evaporation in the lagoon system resulting in increased salinity, and the position of the thermographs on the top of barrier reefs offshore of the main island surface expression, as seen from Google Earth satellite imagery (Figures S1c–S1e), could be some of drivers of the differences between the near-surface (depths <10 m) stations.

The MLR analysis yields an estimate of temperature at each thermograph station that agrees remarkably well with the observed temperatures (Figures 1f, 1g, and S4a–S4c). The MLR accounts for a significant fraction of temperature variability across depths and study sites (Figures S4a–S4c) with depth-averaged R2 values of 0.88, 0.89, and 0.90 at Short Drop Off (Figure S1c), Ulong Rock (Figure S1d), and West Channel (Figure S1e), respectively. The MLR model captures the observed profile changes during different ENSO phases (Figure 1a—dashed lines). As anticipated from Figure S2, SST is the important MLR input at depths in the UML, and SLA becomes important below the MLD. SLA2 can be important near the surface and well below the MLD where the relationship between in situ temperature and SLA is not always linear (Figures S4a and S4b).

Using the MLR-derived coefficients in equation 1 (Figures S4d–S4f), temperature profiles are reconstructed (Figure 2a) back to the first full year of the SST and SLA records (1985, Figures 2b and 2c). Observations of temperature from 2017 were withheld from the MLR model training to validate the model showing good agreement with correlation coefficients between 0.83 and 0.95 for each respective depth and a depth-averaged correlation of 0.90 (Figure S5).

Details are in the caption following the image
(a) Weekly averaged temperatures were reconstructed for Short Drop Off (Figure S1c), on the East coast of Palau, using multiple linear regressions for depths between 0 and 90 m and are shown for the time period of January 1985 to June 2017. Temperature observations from Short Drop Off between January 2000 and December 2016 (white background) were used to train the models. The depth-averaged R2 was 0.88 for the reconstructions during that time period. The models were used to reconstruct temperatures (gray background) in depth as a hindcast (1985–2000) and a forecast (2017). (b) Weekly averaged sea level anomalies observed at the Malakal Harbor gauge (thick black line) are shown with an accompanying linear trend (thin black line). Red and blue patches indicate periods of at least moderate El Niño (ONI > 1) and La Niña (ONI < −1) events, respectively. (c) SSTs at the closest grid point in the CoralTemp 5-km SST product to our observation station at Short Drop Off (thick black line) are shown with an accompanying linear trend (thin black line). (d) The ONI is displayed between 1985 and 2017. ONI = Oceanic Niño Index; SLA = sea level anomaly; SST = sea surface temperature.

The temperature reconstruction shows that the warmest events in the UML at Palau tend to occur during large La Niña conditions that follow moderate to large El Niños, such as during the summers of 1998, 2010, and 2016 following peak El Niño conditions in the preceding winter. The amplitude of the UML warm events tends to increase over the course of the record, reflecting the trend in observed SST of 0.022°C/year between 1985 and 2016 (Figure 2c—thin black line). In addition to high SSTs, these warm events are characterized by downward thermocline displacements corresponding to high SLA conditions.

The warm conditions at Palau following peak El Niños are associated with wind variations north of the equator. SLA variations at Palau are partially driven by local winds and remote winds to the east of Palau. Regionally, the wind-driven response reflects the ENSO zonal dipole pattern and a meridional dipole (Widlansky et al., 2014). SLA variations often reach Palau via westward propagation in the form of Rossby waves. To illustrate this phenomenon, as well as the abrupt warming following El Niños, we examine SST, SLA, and the MLR temperature time-depth variability for 2015–2016 (Figure 3). As the El Niño event weakens, SLA and the surface wind field show a rapid, Pacific-wide change between February and June of 2016 (Figures 3a–3c) with the trade winds extending westward along with a positive SLA. As the wind-forced anomaly reaches Palau in March 2016, local SLA rises (Figure 3g), the thermocline deepens, and UML temperatures abruptly increase (Figure 3h). The anomaly is present through the summer of 2016.

Details are in the caption following the image
The spatial progression of the Rossby wave that shuts down the 2015–2016 El Niño event can be seen in regional SLA, displayed as monthly averages, as the warm pool progressed from the central tropical Pacific in February 2016 (a) westward during the winter and spring of 2016 (b) terminating in the tropical western Pacific in the summer of 2016 (c). Palau is noted in (a)–(c) by a black dot in the western Pacific at 7.5°N, 134°E. Surface level wind fields are overlayed on the SLA subplots (a–c). Monthly averaged SST from (d) February, (e) April, and (f) June show the eastern Pacific and extratropical response to the decay of the 2015–2016 event but no large change around Palau. Vertical black lines in (g) indicate the time stamps of the monthly averages (a–c and d and e). The local response at Palau to this event is evident in the weekly averaged SLA record (g), which is also reflected in reconstructed temperature time series from the east coast of Palau at Short Drop Off (h). Both (g) and (h) are subsets of the records shown in Figures 2b and 2a, respectively. The source data for the SLA (a–c) and SST (d–f) maps are CMEMS SLA 0.25° × 0.25° and NOAA OI SST 0.25° × 0.25°, respectively. Local SLA data (g) are from the NOAA Malakal tide gauge. SLA = sea level anomaly; SST = sea surface temperature; CMEMS = Copernicus Marine Environment Monitoring Service.

The Oceanic Niño Index suggests that a weak La Niña followed the strong El Niño in 2015–2016. Monthly averaged SSTs show high temperatures near Palau and a La Niña-like anomaly by June 2016 (Figures 3d–3f). The maps emphasize that in addition to La Niña conditions, it is the narrow band of easterly wind anomalies around the latitude of Palau (Figures 3b and 3c), which mark the end of strong El Niño events in the spring that lead to the abrupt and long-lasting warming events at Palau.

Based on DHW values computed using SST, potential bleaching events occurred during 1998, 2010, 2014, and 2016 (Figure 4a). Each event occurs after a moderate to large El Niño, except 2014. Extensive bleaching occurred in Palau during the 1998 and 2010 events (Bruno et al., 2001; Golbuu et al., 2007; van Woesik et al., 2012), while minor bleaching occurred during 2014 and 2016 in shallow corals (Colin, personal observation; Text S1). Using the bleaching threshold based on SST, the DHW estimate shows that the warming events at the surface penetrate to 30 m or so, largely reflecting the isothermal conditions in the UML (Figure 4b). Using the scaled bleaching threshold, we find elevated DHW vales extending deeper in the water column to ~50 m, such as during the 1998 La Niña event (Figure 4c). The scaled bleaching thus increases the estimated thermal stress compared to the standard DHW estimate based on only SST (Figure 4c).

Details are in the caption following the image
(a) DHW was computed from the CRW CoralTemp SSTs supplied for Short Drop Off (black line). This matches the record of DHW supplied by CRW (not shown). (b) DHW was computed for the temperature reconstructions between 0 and 90 m, which shows the temporal evolution of the depth variable thermal stress present through the records. DHWs were computed using the MMM supplied for CRW based off of the CoralTemp SST. DHW calculations for (a) and (b) use a bleaching threshold of 1°C. (c) DHWs are shown between 0 and 90 m for the temperature records using the MMM computed for the temperature record at that depth. The bleaching threshold is now the standard deviation of the temperature time series at that depth multiplied by 1°C over the standard deviation of the SST. This gives a proportional variance threshold, which the temperatures would have to exceed to be deemed a thermal stress. All relevant parameters for this figure are in Table S2. DHW = Degree Heating Weeks; CRW = Coral Reef Watch; SST = sea surface temperature; MMM = maximum monthly mean.

4 Conclusions and Discussion

A unique set of long-term, in situ temperature records from Palau provide an opportunity to derive a new empirical model of upper ocean temperature variability at fore reefs using SST and SLA time series. We hypothesize that this relationship could persist throughout the tropical Pacific where the thermocline depth is largely dependent on large-scale wind-driven flows (Widlansky et al., 2014), and SLA has already been shown to be an indicator of the 20°C isotherm depth (Rebert et al., 1985). This concept has been used to derive synthetic temperature profiles in the Atlantic based on satellite altimeter data (Carnes et al., 1990), and temperature and salinity profiles in large-scale ocean models (Fox et al., 2002), adding credence to the hypothesis. We caution that the method does not account for salinity effects and in particular upper ocean thermal structures associated with barrier layers, which are salinity-driven temperature inversions in the tropical western Pacific (Bosc Christelle et al., 2009; Lukas & Lindstrom, 1991). When barrier layers are prevalent, SLA may not always accurately predict the temperature structure.

We conclude that DHW estimates based only on SST may not adequately account for prolonged warming at depth associated with high sea levels and downward thermocline displacements. We have attempted to adjust the bleaching condition with depth to account for observed temperature variations with depth; however, further studies are needed to confirm that the depth-dependent bleaching threshold matches observed bleaching events below the UML where multiple stressors may also play a role in bleaching (Harborne et al., 2017). As a case in point, coral bleaching in Palau was observed at 60-m depth during the 2010 warm event; however, our DHW estimates only show heightened thermal stress to about 30 m, even with the adjusted bleaching condition. Nevertheless, our findings highlight that mesophotic coral reefs are vulnerable to thermal stress and that these depth zones may not be a refugia.

High temporal resolution and long-duration observations of temperature on Palau's fore reef system provide insight into the depth variable structure of thermal stress, leading to a new variance-based approach for predicting thermal stress. Our methods here provide a new means of assessing thermal stress due to the depth-dependent nature of temperature variance seen throughout the range of depths where corals exist in this tropical Pacific location.

Acknowledgments

This work was funded under two Office of Naval Research grants specifically under the Flow Encounter Abrupt Topography DRI. The authors have no competing financial interests to disclose. The authors would like to thank the staff at both the Coastal Research and Development Center at the Scripps Institution of Oceanography and the Coral Reef Research Foundation for making this work possible. Conversations with Megan Cimino, Sophia Merrifield, Mike Fox, and Travis Courtney significantly improved this work. Ganesh Gopalakrishnan and Bruce Cornuelle kindly provided NCEP wind records. Erick Geiger and William Skirving at NOAA's CRW provided site-specific CoralTemp SST output and insights on their methodologies. In situ temperature records can be accessed from the Coral Reef Research Foundation's Water Temperature Catalog at http://wtc.coralreefpalau.org/. All three anonymous reviewers gave insightful comments that greatly improved the manuscript. Example code for implementing the methods presented can be found in the SI or at https://github.com/tschramek/ThermalStress.