Diurnal warming rectification in the tropical Pacific linked to sea surface temperature front

Abstract

Sharp and rapid changes in the sea surface temperature (SST) associated with fronts and the diurnal cycle can drive changes in the atmospheric boundary-layer stability and circulation. Here we show how a one-dimensional surface ocean model forced with either high-resolution or daily averaged surface fluxes can be used to distinguish diurnal versus frontal SST anomalies observed from an uncrewed surface vehicle. The model, forced with daily satellite fluxes, shows that the diurnal warming is largest within the equatorial Pacific cold tongue of SST. The strong persistent SST front north of the cold tongue is evident in both the oceanic and atmospheric boundary-layer stability scales and, as a consequence, in the magnitude of the diurnal ocean warming. Using SST, barometric pressure and surface wind measurements from moorings at 0°, 95° W and 2° N, 95° W, we show that the front in the SST diurnal warming results in a weakened SST front in the afternoon and a corresponding reduced meridional gradient in the barometric pressure that appears to contribute to a diurnal pulsing of the surface meridional winds. To the extent that these modulate the surface branch of the Hadley cell, these diurnal variations may have remote impacts.

Main

Surface wind observations¹ showing cuspy patterns clearly linked to oceanic tropical instability waves (TIWs) demonstrated conclusively that the ocean could force the atmosphere. In the two decades since this seminal study¹, there have been numerous investigations of atmospheric response to sea surface temperature (SST) fronts on a range of scales throughout the global oceans^2,3,4,5, with a lively debate about the mechanisms. Are the wind anomalies caused by the SST front’s influence on atmospheric stability⁶ or the SST front’s impact on a barometric pressure gradient⁷? In a separate line of query, over the past several decades, there has been a growing recognition that the SST diurnal cycle can rectify into longer timescales, impacting air–sea fluxes of heat^8,9, momentum^10,11 and gas^12,13, which in turn can result in large-scale changes to the atmospheric hydrological cycle^14,15,16,17. In this study, we step back and consider how SST fronts affect patterns of diurnal SST warming and how the SST diurnal cycle patterns potentially affect barometric pressure gradients and large-scale circulation.

When winds are weak and solar radiation is large, a diurnal warm layer can form¹² with maximum stratification in late afternoon that is subsequently eroded by night-time cooling and mixing. SST below the diurnal warm layer is referred to as the ‘foundation’ sea surface temperature (T_fdn). In late afternoon, when the diurnal layer is fully formed, T_fdn is found at 10 m or deeper. By contrast, at dawn, after the diurnal warm-layer stratification is completely removed, the foundation SST is the water temperature directly below the thermal skin layer¹⁸. Consequently, the bulk SST measured by an in situ sensor located a few centimetres to a few metres below the surface, T(z_m), can have a large diurnal cycle, with a maximum in late afternoon and a minimum that is equivalent to T_fdn at dawn^12,19,20. Throughout the diurnal cycle, the bulk SST differs from the ocean skin temperature (T_skin). The state-of-the-art Coupled Ocean–Atmosphere Response Experiment (COARE) bulk-flux algorithm^21,22,23,24 thus includes an option to extrapolate the measured bulk in situ SST to T_skin by estimating a warm-layer effect (ΔT_warm) and/or a cool-skin effect (ΔT_cool). The extrapolated skin temperature is then used to compute a more accurate estimate of air–sea fluxes.

In this Article, we show that the Fairall et al.²⁴ diurnal warm-layer model (Fig. 1) used in the COARE bulk-flux algorithm, hereinafter referred to as F96, can also be used to extrapolate the measured bulk SST to T_fdn. For SST and flux time series that cross fronts, extrapolation of the bulk SST to a T_fdn enables separation of SST anomalies due to diurnal warming from anomalies due to fronts. In addition, we show that F96 can be integrated for 24 hours to estimate the daily averaged warm effect from daily averaged surface fluxes if we add a parameterized wind gustiness to the daily averaged wind stress. This then allows us to use daily averaged satellite-based fluxes to examine the spatial patterns of the warm-layer effect that rectify into daily and longer timescales. In places, this rectification can be larger than the nominal cool-skin correction^25,26 of ~0.17 °C. Finally, the model helps explain a curious and consequential feature observed across the observing system^11,27, including from uncrewed surface vehicles (USVs), satellite-based products and moored buoy time series: the sharp SST front in the eastern equatorial Pacific results in a front in the SST diurnal cycle.

Fig. 1: Idealized diurnal warm-layer temperature profile¹⁵.

Trapping depth (D_T) is estimated on the basis of the requirement that bulk Richardson number across the warm layer always exceeds a critical value. The warm-layer effect (∆T_warm) is twice the layer-averaged temperature change (∆T_layer), estimated by the surface forcing of a layer of depth D_T. The bulk sea surface temperature T(z_m) is typically measured at a depth of 0.3–2.0 m; it is extrapolated to a subskin temperature T_subskin through addition of a temperature correction, ∆T_cor, which depends on a proportion of the warm-layer effect. We show that the remaining proportion can be used to extrapolate the bulk temperature to the foundation surface temperature, T_fdn. It is assumed that at dawn the diurnal warm layer is fully mixed so that the measured T(z_m) is equivalent to T_fdn (Methods). Credit: sun icon, Flaticon.com.

Getting to the foundation

By definition, T_fdn is the temperature of the water below the diurnal warm layer. Thus, near-surface SSTs within the diurnal warm layer, such as the 0.6 m SST measured by Saildrone, Inc. USVs during a mission to the eastern (~125° W) tropical Pacific (Fig. 2a,b), show daytime warming with peak values in late afternoon and a minimum SST at dawn (Fig. 2c). By contrast, the extrapolated T_fdn shown in Fig. 2c, computed with the F96 model (Methods and equations (1)–(6)), smoothly tracks dawn SST values. This good agreement with expected characteristics of the foundation SST gives confidence that the model (equation (3)) accurately extrapolates the measured SST to T_fdn.

Fig. 2: TPOS 2017 mission Saildrone #1006 collage.

a, Saildrone #1006 (SD1006) track, coloured by SST measured at 0.6 m. b, TPOS Saildrone. c, Measured SST (black) and extrapolated foundation temperature (T_fdn) (red) for the period 1 October 2017 to 1 January 2018, when the drone was south of 20.23° N plotted as a latitude section along ~125° W (upper panel) and time series (lower panel). d, SST at 0.6 m (black) and extrapolated T_fdn (red) from 10 minute data (top panel) and from daily averaged data with 24 hour running-averaged SST shown in black (second panel). The third panel shows the difference between the SST at 0.6 m and T_fdn computed from 10 minute data (black solid), its 24 hour running average (black dashed) and that computed from daily averaged data (red). The bottom panel shows the daily high-passed SST at 0.6 m (black) and plus and minus values of the diurnal warm-layer effect at 0.6 m estimated from daily averaged data (red). Saildrone measurements and flux time series used in the T_fdn estimate are shown in Extended Data Figs. 1 and 2. Credit: b, Saildrone, Inc.

While daytime warming and night-time cooling can lead to large diurnal SST variability at 0.6 m, sharp fronts and eddies can also cause large sub-diurnal variability in SST if the fronts are moving past the fixed observing platform or if the observing platform is moving across the front. In these cases, it is particularly useful to be able to extrapolate the measured SST to a foundation SST to distinguish SST anomalies due to fronts from SST anomalies caused by diurnal warming. As an example, an abrupt front with T_fdn change greater than 1 °C in 1 km can be identified in the record on 7 December 2017 near 2° N, 125° W (Fig. 2c, top panel) and on 1 February 2018 near 2° N, 125° W (Fig. 2d and Extended Data Fig. 1).

Extrapolated T_fdn is computed for each 10 minute time step from dawn to midnight while the accumulated heat fluxed into the ocean is positive (Methods and equations (3)–(6)). At other times, T_fdn is assumed to be the measured SST—an assumption that is reasonable ~dawn but is not as good at midnight (Fig. 2d). The extrapolated T_fdn often has a discontinuity at midnight, when the warm-layer model integration ends and the T_fdn jumps to the observed bulk SST (Fig. 2d, top panel). This could be caused by biases in the net surface heat flux or wind stress. However, even without errors, terminating the integration at midnight will cause a discontinuity since late-night measured SST can still be within the deepening, weakly stratified warm layer. In these cases, the measured SST at midnight will be warmer than T_fdn. The best estimate of T_fdn from the measured SST will be at dawn when the diurnal cycle reaches its local minimum.

Warm-layer corrections from a 24 hour integration

The midnight discontinuity can be avoided by integrating the diurnal warm-layer model for 24 hours to estimate 24 hour averaged SST warming relative to T_fdn (Fig. 2d). Critically, this daily averaged diurnal warm-layer effect can be estimated from daily averaged fluxes if a parameterized gustiness is included in the daily averaged wind stress (Methods and equations (7)−(9)). The gustiness component accounts for the difference between daily averaged scalar wind-stress magnitude and daily averaged wind-stress vector magnitude and ensures that the daily averaged wind-stress values used to force the warm-layer model are non-zero (Extended Data Fig. 2). Inclusion of a stochastic gustiness has been shown to help performance of surface ocean mixed-layer models forced with daily averaged winds²⁸ and with 3 hour averaged fluxes⁹. Likewise, the COARE bulk-flux algorithms^21,22,23 add a gustiness component to 10 minute and hourly averaged winds to account for missing wind variance at convective timescales. The extrapolated T_fdn based on daily averages matches the predawn SST (Fig. 2d, second panel), and the daily averaged temperature effects estimated from 10 minute and daily averaged fluxes (Fig. 2d, third panel) agree well with mean and RMS differences of, respectively, 0.009 °C and 0.04 °C. These, together with the agreement between the daily averaged temperature effect at 0.6 m and the envelope of diurnal high-passed variability (Fig. 2d, bottom panel), all give confidence that satellite-based daily fluxes can be used to explore patterns in the diurnal warming, such as the reduced SST diurnal cycle amplitude observed as the USV crosses the large-scale front from ~7° N to ~2° N (last two weeks of November 2017 in Fig. 2c).

Effect of SST front on other boundary-layer fronts

With the more complete satellite-based flux fields now available²⁹, we reproduce the wind and SST patterns for the same day (3 September 1999) as the seminal study¹ in Extended Data Fig. 3 and 6 months later in Extended Data Fig. 4. We see that, indeed, turbulent sensible and latent heat fluxes have a cuspy TIW pattern similar to SST, with near-zero turbulent heat fluxes into the cool eastern equatorial Pacific ‘cold tongue’ water and large positive values (warming the atmosphere) to the north in the frontal region (Extended Data Fig. 3). These positive turbulent heat fluxes in the frontal region tend to both cool and destabilize the surface oceanic boundary layer (OBL) and warm and destabilize the surface atmospheric boundary layer (ABL). Consequently, as was hypothesized¹, the SST front acts as a front in the atmospheric Monin–Obukhov stability length scale (Methods), L_atm, with positive and small values (indicating surface stabilization of the ABL) over the equatorial cold tongue and negative values (indicating destabilization) over the warmer frontal region (Fig. 3a,b). As expected on the basis of stability physics, weaker surface winds are found where the ABL is stabilized, and higher winds are found where the ABL is destabilized. Because heat is removed from the ocean and the OBL is destabilized, no SST diurnal warm layer forms in this frontal region (Fig. 3c and Extended Data Figs. 3 and 4).

Fig. 3: J-OFURO3 satellite-based fields for 3 September 1999.

a, Small positive atmospheric Monin–Obukhov stability height scales, indicating regions of stabilized ABL, in metres. b, Small positive oceanic Monin–Obukhov stability depth scales, indicating regions of stabilized OBLs, in metres. c, The 24 hour averaged temperature change in °C due to diurnal warming. SST contours with 1 °C contour intervals are overlaid on all panels. All fields are based on daily J-OFURO3 (ref. ²⁹) fields (Methods).

In the equatorial Pacific cold tongue, net turbulent heat fluxes, combined with enhanced radiative heat fluxes, produce a very strong net surface heat flux into the ocean. This buoyancy flux, together with the weak winds associated with the weakly stratified ABL there, strongly stabilize the equatorial cold tongue’s OBL, as indicated by small and positive oceanic Monin–Obukhov stability length scales, L_oc (Fig. 3b and Extended Data Fig. 3). Not surprisingly, large SST diurnal warming is observed in the cold-tongue region, with daily average values of ~0.8 °C near 0°, 125° W (Fig. 3c) for 3 September 1999. Furthermore, TIW cuspy patterns in the SST front are evident not only in the ABL and OBL stability fronts, but also in the front of the sea surface diurnal warming (Fig. 3c).

Six months later (Extended Data Fig. 4), when the equatorial cold tongue is less well defined, destabilizing turbulent heat fluxes (out of the ocean) are found throughout most of the eastern tropical Pacific, and positive and small L_atm (indicating ABL stabilization) is found only over the coolest equatorial and coastal waters. In general, regardless of season, when there is surface stabilization of the ABL, there tends to be surface stabilization of the OBL. However, the correlation between L_atm and L_oc is not perfect: a stabilizing L_oc is not always associated with a stabilizing L_atm due to the different radiation absorption properties of water and air. Surface radiative fluxes directly affect buoyancy within the OBL but not within the ABL. Thus, ABL and OBL stability length scales are most correlated in cloudy regions, such as in the stratus deck of the coastal upwelling zones, the stratocumulus region of the equatorial cold tongue and the deep convective cumulus regions associated with the intertropical tropical convergence zones (Fig. 3a,b and Extended Data Fig. 3).

Seasonal modulation of the diurnal variations

The Tropical Pacific Observing System (TPOS) mooring array, used for monitoring and predicting El Niño/Southern Oscillation, has been observing hourly or better 1 m SST, thermocline depth and surface meteorology for over 20 years³⁰, enabling analysis of diurnal cycle patterns in both the ocean and atmosphere. In addition, select sites have been enhanced to measure air–sea fluxes and barometric pressure, either as part of a process study^31,32 or as part of the OceanSITES network of sustained observations³³. Seasonal climatologies of ABL stability, indicated by the air–sea temperature difference (Fig. 4), exhibit large variability spatially and over timescales from diurnal and seasonal to the mean. At all sites, with the exception of the eastern and central equatorial Pacific, the ABL is unstable (surface air temperature is cooler than SST) throughout the entire seasonal cycle. The largest unstable air–sea temperature difference is found at 2° N in the eastern Pacific and is probably due to the strong SST front there. While the SST diurnal cycle at 0°, 95° W is relatively large throughout the annual cycle (lower right panel of Fig. 4), at 2° N, 95° W, the SST diurnal cycle is large only during the warm SST season when the SST front is weak.

Fig. 4: SST and air temperature seasonal cycles and diurnal amplitudes for select sites in the TPOS.

Range of the diurnal cycle monthly climatologies (black) are overlaid on the monthly 1 m SST climatology (red) and 3 m air temperature climatology (blue).

As previously found³⁴, the wind-speed diurnal cycle (Extended Data Figs. 5 and 6) is largest where the equatorial cold tongue and its front are most pronounced. The phasing of the meridional wind-speed diurnal cycle at 0°, 95° W (Fig. 5 and Extended Data Fig. 5) is somewhat similar to that found over land. This could suggest surface cooling at night-time causes enhanced stability that shields the surface from higher winds aloft, and warm daytime surface temperatures result in enhanced turbulent mixing that carries high winds to the surface³⁵. Careful inspection, however, shows that other processes besides stability are at play. For one, as previously noted³⁴, during August–November when the meridional wind diurnal cycle is largest at 0°, 95° W (Extended Data Fig. 6), night-time ABL is more, not less, unstable, compared with other months (T_air less than SST; Fig. 4).

Fig. 5: 24-hour climatologies for 1 m SST, barometric pressure and meridional winds in the equatorial band at 95° W.

a, 1 m SST at 0°, 95° W (black) and 2° N, 95° W (red) and their difference: 2° N minus 0° (light blue; axis on right). b, Barometric pressure at 0° N, 95° W (black) and 2°, 95° W (red). c, Equator minus 2° N, 95° W barometric pressure difference in units hPa. d, Meridional wind anomalies at 0°, 95° W (black), 2°N, 95°W (red), and computed from the pressure gradient anomaly (light blue) in m s⁻¹. All 24 hour climatologies are shown in local time and were computed for the subset of days in the August–November months for the years 2000–2003 when winds, SST and barometric pressure were measured at both sites. In c, the corresponding acceleration caused by the pressure gradient anomaly is also shown in units m s⁻¹ d^–1 on the left axis. Standard errors are shown with dashed lines and assume that each day in the composite is an independent degree of freedom.

Moored barometric pressure time series at 0°, 95° W and 2° N, 95° W (Fig. 5b,c) show that while the semi-diurnal cycle in barometric pressure dominates at each of these sites, the diurnal cycle dominates in the meridional pressure gradient and is roughly of the correct magnitude (although 25% too large) and phasing to account for the diurnal pulsing in the surface meridional winds, as previously speculated³⁴. If the diurnal pulsing of the SST front (Fig. 5a and Extended Data Fig. 7) was fully responsible for these diurnal pressure variations and resulting surface meridional wind pulsing, the barometric pressure sensitivity to SST variations would need to be roughly −0.4 hPa K⁻¹, which is over three times larger than the pressure gradient response to SST front variations associated with TIW³¹. This large apparent sensitivity suggests that other correlated processes might also be at play, such as differential heating of the troposphere across a mean cloudiness and moisture front³⁶.

Conclusion

There is growing recognition that accurate estimation of air–sea exchanges of heat, moisture, momentum and gases depend on proper representation of the ocean’s skin temperature, which is generally cooler but can occasionally be warmer than the bulk SST measured by in situ sensors^8,9,24,25,26. When computing air–sea fluxes with the COARE bulk-flux algorithm, measured SST is extrapolated to the surface using a cool-skin correction and a warm-layer correction estimated from a simple one-dimensional mixed-layer model²⁴ (Fig. 1). In this Article, we show that the F96 warm-layer model can be used to also extrapolate SST to a foundation SST below the diurnal warm layer. Furthermore, using independent observations from the TPOS, including from satellites, buoys and USV, we show that the F96 model integrated for 24 hours can be used to estimate the daily averaged warm-layer effect from daily averaged fluxes if a parameterized gustiness is added to the daily averaged wind stress. This gustiness component is equivalent to the sub-diurnal relative wind-speed variance, or equivalently to the difference between the daily averaged wind-stress magnitude and the magnitude of the daily averaged wind-stress vector (Extended Data Fig. 8). We use a simple parameterization of the sub-diurnal gustiness variance that is based on SST³⁷ (Methods) since gustiness will be largest in unstable ABL conditions and the ABL’s stability is correlated with SST (Fig. 3). We also set a minimum value for wind stress, below which we assume gustiness prevails (Extended Data Fig. 8). Developing and testing more sophisticated parameterizations of the diurnal gustiness will be the subject of future studies.

Using satellite-based air–sea fluxes²⁹ to revisit the seminal 3 September 1999 case study¹, it is shown that the eastern equatorial Pacific cold tongue’s SST front is also a front in the SST diurnal warming, with a large SST diurnal cycle that rectifies into the daily averaged SST in the equatorial cold tongue. By contrast, SST diurnal cycle is minimal in the warmer frontal regions flanking the equatorial cold tongue. This SST diurnal cycle front represents a front in the OBL and ABL stability length scales, with stabilized OBL and ABL where the SST diurnal cycle is large and destabilized OBL and ABL where the SST diurnal cycle is minimized. We show that the strong diurnal warming on the cold side of the SST front leads to a diurnal cycle in the strength of the SST front (strongest at dawn and weakest in late afternoon), which appears to lead to a diurnal pulsing of the barometric pressure gradient and meridional surface winds. To the extent that surface heating (as opposed to direct heating of the atmosphere) plays a role in the ABL response to the diurnal (S1) thermal tide, one might expect these SST frontal patterns to modulate not only the diurnal pulsing of the surface branch of the Hadley cell, but also potentially other aspects of the large-scale atmospheric circulation. The novel USVs fill an important gap within the TPOS, providing high temporal- and high spatial-resolution observations over large regions.

The diurnal cycle is a fundamental timescale, representing an external forcing of Earth’s system. Daytime near-surface ocean stratification can trap momentum and other properties, leading to rectified impacts on the ocean^{10,11,38,39,40}. Because convective mixing is organized at this timescale, it represents a critical building block process for large-scale convection. By understanding the patterns of the diurnal cycle, we can understand the large-scale patterns stabilizing and destabilizing the OBL and ABL and better understand patterns of convection over the oceans. As such, the diurnal cycle acts as an excellent metric for testing physics of numerical general circulation models.

Methods

Saildrone air–sea flux observations

Saildrones are wind- and solar-powered USVs manufactured and piloted by Saildrone, Inc. Data used here are from drone 1006 during the 2017 TPOS mission⁴¹ to the tropics at 125° W (Fig. 2 and Extended Data Figs. 1 and 2). Measurements include wind speed and direction at 5 m height in Earth coordinates, air temperature and humidity at 2.4 m, barometric pressure at 0.3 m, incoming solar radiation and incoming long-wave radiation, SST at 0.6 m and surface currents at 6 m. Air–sea fluxes were computed from 10 minute averages using the COARE²³ v.3.5 bulk algorithm. Intercomparisons against a moored flux buoy⁴¹ suggest Saildrone net surface heat flux and wind-stress errors are comparable to the flux mooring⁴² (10 W m⁻² and 0.007 N m⁻²).

Satellite-based air–sea flux observations

Japanese Ocean Flux datasets with Use of Remote Sensing Observations version 3 (J-OFURO3) is a global dataset of air–sea heat, freshwater and momentum fluxes and related parameters (for example, SST) derived from satellite-based observations²⁹. The third-generation product is available as daily means on a 0.25° grid for 1988–2017. It should be noted that the nominal depth of the SST in the J-OFURO3 product is ambiguous as it depends on the treatment of SST from multiple products. Its reliance on microwave satellite products, however, suggests that the SST is probably closer to a subskin temperature rather than a foundation temperature. Its net surface heat flux had a mean bias relative to global buoys of −5.8 W m⁻² (excessive ocean warming)⁴³.

Climatological diurnal cycles in the tropical Pacific

High-resolution 10 minute SST at 1 m, solar radiation and wind-speed data from National Oceanic and Atmospheric Administration’s moored buoy array within the TPOS²⁵ over the period of 1998–2019 are used to create monthly, seasonal (3 month averaged) composites of the diurnal cycle. Barometric pressure on and off the Equator was available only during a process study³¹ from 2000 to 2003, thus limiting the climatological period for variables shown in Fig. 5. When the zenith angle is large (as the Sun rises and sets), the shields on the air temperature sensors are not necessarily effective at blocking beams of sunlight from directly warming the sensor^44,45. Consequently, while we show the air temperature diurnal peaks in Fig. 4, in this study we do not compute the diurnal cycle in the ABL stability as represented by the SST minus surface air temperature difference.

As a first step towards creating the diurnal cycle composites, the 10 minute data are hourly averaged, filtered with a 3 day high-pass filter, and then binned in 24 hours of the day. The diurnal cycle is calculated by averaging the data in each hourly bin with its uncertainty measured by the standard error with the assumption that each day is an independent degree of freedom. Seasonal climatologies were created by composite analysis of daily data.

SST definitions

SST has a large vertical structure near the air–sea interface¹⁸. The ocean skin temperature (T_skin) felt by the atmosphere is nearly always cooler than the water just a millimetre below the surface^21,25,46,47. By contrast, solar radiation can penetrate the thermal skin layer, forming a diurnal warm layer when winds and air–sea turbulent heat fluxes are weak. Thus, bulk SST is typically referred to with its measurement depth, that is, 1 m SST. The foundation SST (T_fdn) is the SST below the diurnal warm layer. Estimates of T_skin from measurements of T_fdn require correction for the cool-skin effect with magnitude ΔT_cool and a possible adjustment for the warm-layer effect (ΔT_warm):

$${T}_{{{\mathrm{skin}}}}={T}_{{{\mathrm{fdn}}}}+\Delta {T}_{{{\mathrm{warm}}}}-\Delta {T}_{{{\mathrm{cool}}}}$$

(1)

F96 warm-layer model for computing foundation temperature

Because skin temperature is a key state variable for air–sea turbulent (latent and sensible) heat fluxes, the COARE bulk-flux algorithm^21,22,23 includes a cool-skin correction and a warm-layer correction²⁴ based on a simplified one-dimensional mixed-layer model⁴⁸, forced by the air–sea fluxes estimated using the measured bulk SST. Night-time mixing is assumed to cause the bulk SST to be identical to the foundation SST just before dawn. The diurnal warm-layer stratification is also assumed to be uniform above a trapping depth (D_T) so that the ΔT_warm is twice the layer-averaged temperature change relative to T_fdn measured by the bulk SST just before dawn at ~6:00 lt (Fig. 1). The T_skin can then be estimated from a measured bulk SST at depth z_m according to:

$${T}_{{{\mathrm{skin}}}}=T({z}_{{\mathrm{m}}})+\Delta {T}_{{{\mathrm{warm}}}}\frac{{z}_{{\mathrm{m}}}}{{D}_{{\mathrm{T}}}}-\Delta {T}_{{{\mathrm{cool}}}}$$

(2)

The second term on the right-hand side of equation (2) is often referred to as the warm-layer correction (for example, Fig. 1), while the third term is the cool-skin correction (typically 0.17 °C) (refs. ^25,26).

For computation of the foundation temperature based on the measured bulk SST, equations (1) and (2) can be rearranged according to:

$${T}_{{{\mathrm{fdn}}}}=T({z}_{{\mathrm{m}}})-\Delta {T}_{{{\mathrm{warm}}}}+\Delta {T}_{{{\mathrm{warm}}}}\frac{{z}_{{\mathrm{m}}}}{{D}_{{\mathrm{T}}}}$$

(3)

Note that the third term on the right-hand side of equation (3) is the warm-layer correction for skin temperature extrapolations and contributes only if the trapping depth is relatively shallow (not more than twice as deep as the measurement depth). The T_fdn in equation (3) can thus be calculated for each time step and saved during runtime of the warm-layer correction of the COARE bulk-flux algorithm^21,22,23.

The F96 warm-layer model modified for 24 hour integration

Because a linear temperature profile above the trapping depth is assumed, the warm-layer temperature change at the air–sea interface will be double the layer-averaged temperature change, that is, ΔT_warm = 2ΔT_layer. As shown in Fig. 1, F96 (ref. ²⁴) estimates the vertically averaged temperature and velocity changes within a layer above a trapping depth, D_T, which is defined as where the surface-forced conditions relative to the foundation temperature reach a critical Richardson number of 0.65. In particular:

$$\Delta {T}_{{{\mathrm{layer}}}}(t)=\frac{1}{{\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}}{D}_{{\mathrm{T}}}}{\int }_{6{{\mathrm{LT}}}}^{t}(Q_{0}-{Q}_{{{\mathrm{pen}}}})\partial t\,=\frac{\overline{{{Q}}_{0}}-\overline{{{Q}}_{{{\mathrm{pen}}}}}}{{\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}}{D}_{{\mathrm{T}}}}{\int }_{6{{\mathrm{LT}}}}^{t}\partial t$$

(4)

$$\Delta {u}_{{{\mathrm{layer}}}}(t)=\frac{1}{{D}_{{\mathrm{T}}}}{\int }_{6{{\mathrm{LT}}}}^{t}{u}_{* ,{{\mathrm{oc}}}}^{2}\partial t=\frac{1}{{D}_{{\mathrm{T}}}}({\overline{{u}_{* ,{{\mathrm{oc}}}}}}^{\,2}+\overline{{{{\mathrm{gust}}}}_{{{\mathrm{oc}}}}}){\int }_{6{{\mathrm{LT}}}}^{t}\partial t$$

(5)

$${{Ri}}_{{cr}}=\frac{{g\frac{\partial \rho }{\partial T}\Delta T}_{\mathrm{layer}}{D}_{\mathrm{T}}}{2{\rho }_{\mathrm{oc}}\,{\Delta {u}_{\mathrm{layer}}}^{2}}=0.65$$

where ρ_oc is surface ocean density, C_p,oc is the heat capacity of water, g is gravity, the overbars indicate an average from 6:00 lt to time t, Q₀ is the net surface heat flux into the ocean surface, ${u}_{* ,{{\mathrm{oc}}}}$ is the oceanic frictional velocity estimated from the wind stress and $\overline{{{{\mathrm{gust}}}}_{{{\mathrm{oc}}}}}$ is the variance (‘gustiness’) in ${u}_{* ,{{\mathrm{oc}}}}$ (see the following). Penetrative radiation, Q_pen, is estimated from the net solar radiation (Q_sw,net, that is, incoming minus reflected solar radiation) and an absorption profile, f, evaluated at the trapping depth:

$$\begin{array}{c}{Q}_{{\mathrm{pen}}}(z={D}_{{\mathrm{T}}})={Q}_{{\mathrm{sw,net}}}\,f({D}_{{\mathrm{T}}})\\ f=\left(\right.\!0.004(1\,\mbox{--}\,{\mathrm{exp}}(\mbox{--}{D}_{{\mathrm{T}}}/0.014))+0.096(1\,\mbox{--}\,{\mathrm{exp}}(\mbox{--}{D}_{{\mathrm{T}}}/0.36))\\ +\,[5.77(1-{\mathrm{exp}}(\mbox{--}{D}_{{\mathrm{T}}}/12.82))]\!\left.\right)/{D}_{{\mathrm{T}}}\end{array}$$

We list here the full absorption polynomial used in the latest version of the COARE bulk algorithm’s warm-layer model^49,50, although typically for the 24 hour integration, only the term within the square brackets is significant.

Using equations (4) and (5) within the Richardson number and defining the thermal expansion coefficient as α = –(1/ρ_oc)(∂ρ/∂T), the trapping depth at time t can then be expressed as:

$${D}_{{\mathrm{T}}}={\left[2{{{{Ri}}}}_{{{{cr}}}}{\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}}\right]}^{1/2}\frac{{\int }_{6{{\mathrm{LT}}}}^{t}{u}_{* ,{{\mathrm{oc}}}}^{2}\partial t}{{\left[\alpha g{\int }_{6{{\mathrm{LT}}}}^{t}(Q_{0}-{Q}_{{{\mathrm{pen}}}})\partial t\right]}^{1/2}}$$

(6)

$${D}_{{\mathrm{T}}}={\left[\frac{2{{{{Ri}}}}_{{{{cr}}}}{\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}}I}{\alpha g\left(\overline{{{{Q}}}_{0}}-\overline{{{{Q}}}_{{{\mathrm{pen}}}}}\right)}\right]}^{\frac{1}{2}}\left({\overline{{u}_{* ,{{\mathrm{oc}}}}}}^{\,2}+\overline{{{{\mathrm{gust}}}}_{{{\mathrm{oc}}}}}\right)$$

(7)

To ensure that ΔT_warm is positive definite, the model integrations in equations (4)–(7) are performed only when heat flux is stabilizing (that is, $\overline{{{{Q}}}_{0}}$ > 0 W m⁻²). At other periods, it is set to zero and the bulk SST is assumed to be the foundation SST.

Within the COARE bulk algorithm, the F96 diurnal warm-layer model (equations (4)–(6)) evaluates the integral as a summation of the high-resolution (for example, 10 minute) integrand. We note, however, that this is equivalent to its average multiplied by the averaging period in units seconds, as long as a gustiness parameter is added to the averaged frictional velocity or wind stress. Thus, while Fig. 2c shows foundation temperature extrapolations computed at each 10 minute time step, in Figs. 2d and 3, the integration is extended to 24 hours (that is, $I={\int }_{0}^{t}\partial {t}$ = 86,400 seconds, and the overbars indicate daily averages). To estimate $\overline{{{{Q}}}_{{{\mathrm{pen}}}}}$ initially, an empirical relation for D_T as a function of the Monin–Obukhov depth, L_oc, is used in equation (7). A first estimate of the trapping depth is then estimated by equation (7) and used to recompute the Q_pen, which then is used to compute the trapping depth according to equation (7).

With the trapping depth then determined, the warm-layer effect can be estimated as:

$$\Delta {T}_{{{\mathrm{warm}}}}=A\frac{{\left(\overline{{{{Q}}}_{0}}-\overline{{{{Q}}}_{{{\mathrm{pen}}}}}\right)}^{3/2}}{\left({\overline{{u}_{* ,{{\mathrm{oc}}}}}}^{\,2}+\overline{{{{\mathrm{gust}}}}_{{{\mathrm{oc}}}}}\right)},$$

(8)

where $A={\left[(2\alpha {gI})/({({\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}})}^{3}{{{{Ri}}}}_{{{{cr}}}})\right]}^{1/2}$.

The foundation temperature can then be estimated from the measured bulk temperature from equation (3). As shown in Fig. 2d, this net warming for the 24 hour period, computed from daily averaged fluxes, matches the 24 hour averaged high-resolution warming estimate as well as the envelope of high-passed sub-diurnal variability measured by the 0.6 m thermistor, giving confidence that equation (8) can be used with global daily averaged flux fields to analyse patterns in diurnal warming. Furthermore, the J-OFURO3 product is able to qualitatively reproduce the 1 m SST diurnal cycle measured by the TPOS buoy at 0°, 95° W (Extended Data Fig. 8), with discrepancies probably associated with sampling differences and errors in the satellite-based fluxes. Propagation of errors in equation (8) shows that for a 150 W m⁻² net heat flux into the ocean trapping layer with 3 m s⁻¹ wind speed, a 10 W m⁻² error would result in a 10% (0.07 °C) error in the warm-layer effect. However, for these low wind-stress conditions, a small 0.007 N m⁻² error can account for a 62% error in wind stress and a corresponding 62% (0.34 °C) error in the warm-layer effect ΔT_warm.

Diurnal gustiness

In the F96 model, a convective gustiness is applied to the wind velocity measurements at each time step of the integration. In our analytical version for the 24 hour integration, a diurnal gustiness must be added to the winds to account for diurnal and sub-diurnal variability in the wind speed and direction. This is particularly important in the equatorial cold-tongue region where winds are very light and currents are strong, without which division by zero led to unrealistically large diurnal warming (>20 °C warming). We therefore use a diurnal wind-speed gustiness formula that depends on SST³⁷:

$$\begin{array}{c}\Delta {\rm{ws}}=0.18\,{\rm{m}}\,{\rm{s}}^{-1}{\,}^\circ {\rm{C}}^{-1}({\rm{SST}}-18.1{\,}^\circ {\rm{C}})\,{\rm{for}}\,23.7^\circ {\rm{C}} < {\rm{SST}} < 29.8^\circ {\rm{C}}\\ \Delta {\rm{ws}}=1\,{\rm{m}}\,{{\rm{s}}}^{-1}{\rm{for}}\,{\rm{SST}} < 23.7{\,}^\circ {\rm{C}}\\ \Delta {\rm{ws}}=2.1\,{\rm{m}}\,{{\rm{s}}}^{-1}{\rm{for}}\,{\rm{SST}} > 29.8{\,}^\circ {\rm{C}}\end{array}$$

When this wind-speed gustiness, in units metres per second, is combined with a nominal drag coefficient of 0.001, the frictional velocity gustiness can be estimated as:

$$\overline{{{{\mathrm{gust}}}}_{{{\mathrm{oc}}}}}=\frac{{\rho }_{{\mathrm{a}}}}{{\rho }_{{{\mathrm{oc}}}}}\overline{{{{\mathrm{gust}}}}_{{{\mathrm{at}}}}}=\frac{{\rho }_{{\mathrm{a}}}}{{\rho }_{{{\mathrm{oc}}}}}0.001\times {\Delta {{\mathrm{ws}}}}^{2}$$

(9)

Finally, to match the minimum wind stress observed in the equatorial cold tongue at 0°, 95° W when computed from winds relative to surface currents, the gustiness was increased so that the daily averaged wind stress enhanced by the atmospheric gustiness ($\overline{{\rm{gust}}_{\rm{at}}}$) never fell below 0.01 N m⁻², roughly equivalent to a wind speed of 2.8 m s⁻¹.

Atmospheric and oceanic stability length scales

Air–sea heat and moisture fluxes that affect the density stratification act as a surface buoyancy flux. The net surface heat flux at the ocean surface (Q₀), including the radiative fluxes at the surface of the ocean, and the net precipitation (P) minus evaporation (E) are included in the oceanic buoyancy flux (B_0,oc). By contrast, the surface ABL stratification is relatively insensitive to radiative fluxes and precipitation, and thus the atmospheric buoyancy flux (B_0,atm) depends simply on the surface virtual temperature flux $\overline{{w^{\prime}}{T}_{{\mathrm{v}}}^{\prime} }{|}_{0}$, which can be related to the sensible (Q_sen) and latent (Q_lat) heat fluxes according to:

$${B}_{0,{{\mathrm{atm}}}}=\frac{g}{{T}_{\mathrm{v}}}\overline{w^{\prime} {T}_{{\mathrm{v}}}^{\prime} }{{\rm{|}}}_{0}=\frac{g}{{T}_{{\mathrm{v}}}}\left(\frac{{Q}_{{{\mathrm{sen}}}}}{{\rho }_{{{\mathrm{atm}}}}{C}_{{\mathrm{p}},{{\mathrm{atm}}}}}+0.61\frac{{Q}_{{{\mathrm{lat}}}}}{{\rho }_{{{\mathrm{atm}}}}{L}_{{\mathrm{e}}}}\right)$$

$${B}_{0,{{\mathrm{oc}}}}=\frac{g}{{\rho }_{{{\mathrm{oc}}}}}\,\overline{w^{\prime} {\rho }_{{{\mathrm{oc}}}}^{\prime} }{{\rm{|}}}_{0}=\frac{g}{{\rho }_{{{\mathrm{oc}}}}} \left( \frac{\partial \rho }{\partial T}\,\frac{{Q}_{0}}{{\rho }_{{{\mathrm{oc}}}}{C}_{{\mathrm{p}},{{\mathrm{oc}}}}}+\frac{\partial \rho }{\partial S}S(P-E)\right)$$

where g is gravity, S is the salinity of the surface water, T and T_v are water temperature and virtual air temperature, and C_p,atm and C_p,oc are, respectively, the specific heat of air and water.

The Monin–Obukhov stability scale is the height where the wind-stress production of mechanical turbulent energy balances or is equivalent to the buoyancy flux. The stability scales for the atmospheric (L_atm) and oceanic (L_oc) boundary layers are computed using, respectively, the atmospheric buoyancy flux (B_0,atm) and oceanic buoyancy flux (B_0,oc):

$${L}_{{{\mathrm{atm}}}}=z\left(\overline{u{^\prime} w{^\prime} }{{\rm{|}}}_{0,{{\mathrm{atm}}}}\,\partial {u}_{{{\mathrm{atm}}}}/\partial z\right)/{B}_{0,{{\mathrm{atm}}}}={u}_{* ,{{\mathrm{atm}}}}^{3}/\left(k{B}_{0,{{\mathrm{atm}}}}\right)$$

$${L}_{{{\mathrm{oc}}}}=z\left(\overline{u{^\prime} w{^\prime} }{{\rm{|}}}_{0,{{\mathrm{oc}}}}\,\partial {u}_{{{\mathrm{oc}}}}/\partial z\right)/{B}_{0,{{\mathrm{oc}}}}={u}_{* ,{{\mathrm{oc}}}}^{3}/\left(k{B}_{0,{{\mathrm{oc}}}}\right)$$

where k is the von Karmon constant (taken to be 0.4 for both the atmosphere and ocean, with $\partial u/\partial {z}\sim {u}_{* }/({kz})$). The frictional velocities in the oceanic (${u}_{* ,{{\mathrm{oc}}}}$) and atmospheric (${u}_{* ,{{\mathrm{atm}}}}$) boundary layers of density ρ_oc and ρ_atm relate to the wind stress (${\tau }_{0}$) according to:

$${\tau }_{0}={\rho }_{{{\mathrm{atm}}}}\overline{u{^\prime} w{^\prime} }{{\rm{|}}}_{0,{{\mathrm{atm}}}}={\rho }_{{{\mathrm{atm}}}}\,{u}_{* ,{{\mathrm{atm}}}}^{2}={\rho }_{{{\mathrm{oc}}}}\,{u}_{* ,{{\mathrm{oc}}}}^{2}={\rho }_{{{\mathrm{oc}}}}\overline{u{^\prime} w{^\prime} }{{\rm{|}}}_{0,{{\mathrm{oc}}}}$$

where $\overline{{u{^\prime}w{^\prime}}}{|}_{0,{{\mathrm{atm}}}}$ and $\,\overline{{u{^\prime}w{^\prime}}}{|}_{0,{oc}}$ are the co-varying horizontal and vertical turbulent motions in the surface layer of, respectively, the ABL and OBL. Because L_atm and L_oc in Fig. 3 are estimated with daily averaged data, the frictional velocities include the gustiness component (equation (9)).

Our sign convention is intended to indicate magnitudes that are positive throughout most of the ocean. Thus, a positive value for net surface heat flux and net solar radiation warm the ocean, while net infrared radiation and turbulent sensible and latent heat fluxes are positive if they cool the ocean:

$${Q}_{0}={Q}_{{{\mathrm{sw}}},{{\mathrm{net}}}}-{Q}_{{{\mathrm{lw}}},{{\mathrm{net}}}}-{Q}_{{{\mathrm{sen}}}}-{Q}_{{{\mathrm{lat}}}}$$

Thus, in most parts of the world’s ocean where air temperature is cooler than the ocean skin temperature, Q_sen would have a positive value, indicating turbulent heat fluxes that warm and destabilize the ABL and cool and destabilize the OBL. Likewise, in regions where the turbulent air–sea heat flux warms the OBL and cools the ABL, the atmospheric and oceanic buoyancy fluxes will tend to stabilize the corresponding boundary layers. The ocean can also be stabilized by the net radiative heat fluxes, while the atmosphere tends to be transparent to the radiative fluxes and therefore the atmospheric buoyancy flux depends only on the turbulent fluxes affecting the virtual temperature.