VIKING20X (short: V20) is a global ocean general circulation model configuration based on the NEMO code version 3.6; at 1/4° horizontal resolution with an embedded 1/20° nest in the Atlantic . This translates into a grid spacing of about 3 km in the North Sea. The nest domain includes the entire North Sea and large parts of the Baltic Sea. On both grids, 46 z-levels are used. The grid cell thickness increases from 6 m at the surface to 250 m in the deep ocean. The model configuration has been successfully used to study MHWs on the Canadian Shelf and throughout the entire Atlantic .

The experiment analysed here VIKING20X.L46-KTS001; follows the experimental design recommended by the OMIP-II protocol for the first model cycle . Accordingly, the experiment is initialized from rest and temperature and salinity from the World Ocean Atlas 2013 climatology . The surface heat and momentum fluxes are derived from reanalyzed atmospheric states provided by the JRA55-do v1.4 dataset using bulk formulae. The namelist settings follow the “VIKING20X-JRA-OMIP” experiment described in with a few differences that were applied to further improve the representation of the North Sea in the model. The most important differences are a new bathymetry with a more realistic coastline in the nest domain (in particular in the North Sea) and a different distribution of runoff. In VIKING20X.L46-KTS001, runoff is not only added to the uppermost grid cell, but more realistically distributed over the water column. The depth range over which the runoff is distributed depends on the climatological strength of the runoff and is about 12 m for rivers entering the North Sea. For rivers outside the North Sea it can be up to 50 m. Additionally, the surface air pressure is used as a forcing and directly impacts the dynamics of the ocean. The simulations based on VIKING20X do not include tidal forcing. The comparison of different datasets described below, as well as additional experiments with and without tides in a model with lower horizontal resolution (not shown), suggest only a small impact of tides on the characteristics of surface MHWs in our focus region (German Bight).

To study the sensitivity of the North Sea to remote temperature variability in the Atlantic we conducted a 10-year long sensitivity experiment (VIKING20X.L46-KTS002). In this experiment the heat flux inside and outside the North Sea are derived from different decades. The experiment branches off from the reference experiment (VIKING20X.L46-KTS001) in 2010. Within the North Sea, heat and momentum fluxes are derived from the atmospheric state of the years 2010 to 2019. Therefore, the atmospheric state in the sensitivity and reference experiments are exactly the same, but the heat and momentum fluxes may differ as they depend on the ocean state. Outside the North Sea, the momentum flux is derived from the same period (2010–2019), but the surface heat flux is derived from the atmospheric state starting in 1990 and ending 1999. The experiment is compared to the years 2010–2019 in the reference experiment. This allows us to isolate the effect of changing ocean temperature outside the North Sea, while maintaining atmospheric conditions inside the North Sea, including the wind driven transport across the North Sea's boundary.

Although VIKING20X is a highly capable model for simulating the large-scale Atlantic circulation, it was not optimised for simulating the dynamics of marginal seas. To compare our results with a forced model that was specifically designed to simulate the North Sea circulation, we use output from the HIROMB-BOOS-Model (HBM) based operational model of the Federal Maritime and Hydrographic Agency BSH;. The data (short name: BSH) was obtained from https://data.bsh.de/OpenData/OperationalModel/ (last access: 10 January 2025). This regional model has a horizontal resolution of 0.9 km in the German Bight and 5 km in the remaining North Sea. The model uses 24 vertical levels of 2–5 m thickness. The grid cell thickness is fixed, but their centre with respect to a fixed depth axis varies temporally with sea surface height. The temperature is provided after interpolation onto such a fixed depth axis. Additionally the value of the uppermost model level was used without interpolation as the “surface” temperature. The model is forced by operational forecasts provided by the German Weather Service (DWD). Due to the use of tidal forcing at the open boundaries of the regional model and a higher horizontal and vertical resolution, which is also connected to a more realistic bathymetry, the BSH model is expected to simulate the physical conditions in the German Bight more realistically than VIKING20X. However, the model data is only available after 2016. Therefore, no MHW statistics (that are based on a 30-year baseline) can be derived from this dataset and it was only used to validate the vertical structure of MHW events that occurred after 2016.

Similar to the BSH model the coastDat1 dataset (short: CST1) was created with a focus on the North Sea, including tides. The data was obtained from the World Data Center for Climate (10.1594/WDCC/coastDat-1_HAMSOM, ). It is based on the HAMburg Shelf Ocean Model (HAMSOM) with a 20 by 20 km horizontal resolution and 19 vertical levels with a vertical resolution of 5 m at the surface . Thus the horizontal resolution is lower compared to VIKING20X and the BSH model. The vertical resolution is similar to VIKING20X, but lower compared to the BSH model. HAMSOM is a regional model of the North Sea and lateral boundary boundary conditions were obtained from a coarse-grid model of the Northwest-European Shelf Sea. The model was forced by 6-hourly NCEP/NCAR reanalysis fields. The available data ranges from 1948 to 2007. Only the common time period compared with the other datasets (1980–2007) is used here.

In contrast to the previous three model configurations that only incorporate atmospheric observations (via atmospheric reanalysis or initialized forecasts), the GLORYS12 version 1 reanalysis (short G12) additionally assimilates ocean observations . In particular, this includes SST from satellites. The data was obtained from the E.U. Copernicus Marine Service (10.48670/moi-00021, ). GLORYS12 is a NEMO-based model with a global 1/12° resolution (about 9 km in the North Sea) and 50 vertical levels. The vertical spacing is 1 m at the surface and increases with depth (22 levels are within the first 100 m). In addition to satellite-based SST measurements, GLORYS12 also assimilates temperature and salinity profiles from the Argo database. GLORYS12 uses a reduced-order Kalman filter approach for small-scale corrections towards observations and a 3D variational (3D-Var) approach to correct larger scales . These approaches have major advantages compared to a simple nudging for example, but they can still introduce spurious sources of heat. The atmospheric forcing is provided by the ECMWF ERA-Interim reanalysis. As VIKING20X, GLORYS12 does not include tidal forcing (although the assimilation of observations implicitly introduces the effect of tides in GLORYS12, especially close to the surface). The dataset covers the years from 1993 to 2023.

Similar to the comparison between VIKING20X and the BSH model, the GLORYS12 reanalysis is a global model and was not designed with the North Sea in focus. The European Northwest Shelf reanalysis (short: NWS) on the other hand was specifically set-up to simulate the dynamics of the Northwest European Shelf. The data was obtained from the E.U. Copernicus Marine Service (10.48670/moi-00059, ). It is based on the NEMO ocean model and assimilates observations, including SST from satellite measurements and vertical temperature profiles from various data sources. In contrast to GLORYS12, only a variational assimilation scheme, NEMOvar , is used. The atmospheric forcing is provided by the ECMWF ERA5 atmospheric reanalysis. In contrast to the previous model set-ups that use z-levels in the vertical, the NWS reanalysis uses 51 hybrid s-sigma terrain-following coordinates. The data is provided after interpolation onto a regular grid with 24 depth levels. The surface value is not interpolated and based on a layer of less than 1 m thickness. Furthermore, the model incorporates tidal forcing at the open boundaries and the equilibrium tide from a tidal model. The dataset covers the years from 1993 to 2023.

A dataset that does not involve a numerical model is the OSTIA (short: OST) product created by the UK MetOffice . Data was obtained from 10.48670/moi-00168 (1982–2021) and 10.48670/moi-00165 (2022–2023). OSTIA provides processing level 4 satellite observations. It uses SST derived from Advanced High Resolution Radiometer (AVHRR) and microwave radiometer measurements. In addition to satellite measurements, in-situ temperature observations from buoys, for example, are used. The different data sources are blended using the NEMOvar scheme. Although NEMOvar is often used together with the NEMO ocean model (e.g. NWS), this dataset does not involve the numerical ocean model step and is therefore distinct from the ocean reanalysis datasets. The dataset is provided on a 1/20° regular grid (about 5 km in the North Sea) and covers the years from 1982 to 2023.

Similarly, the NOAA OISST version 2.1 (short: OIS) uses satellite based AVHRR measurements together with in-situ measurements from various platforms, including ships and Argo floats . Data was obtained from https://www.ncei.noaa.gov/data/sea-surface-temperature-optimum-interpolation/v2.1/access/avhrr/ (last access: 30 October 2024). In contrast to OSTIA, an optimal interpolation scheme is used to create a gap-free dataset instead of the NEMOvar scheme. The data is provided on a regular 1/4° grid (about 20 km in the North Sea) and covers the years from 1982 to 2023.

Another level 4 processing dataset is provided by the Danish Meteorological Institute based on the DMIOI system . The data was obtained from 10.48670/moi-00156, . The DMI dataset (short: DMI) is based on satellite observations from different infrared and microwave sensors and interpolated using an optimal interpolation scheme. Key differences to the previously mentioned datasets are that no additional in-situ data is used and the very high horizontal resolution of the dataset (1/50° or about 2 km in the North Sea). The dataset covers the years from 1982 to 2023.

Marine heatwaves are defined following the definition of . For each day of the year, the climatological mean and 90th percentile of all temperature values within an 11 d window are estimated. The climatology and 90th percentile are then smoothed using a 31 d moving average. A MHW is present if the temperature exceeds the seasonally varying 90th percentile for at least five consecutive days. If the gap between two MHWs does not exceed 2 d, they are considered a single event. This definition was applied separately to all grid points of the datasets.

While the definition is accepted by almost all researchers in the MHW community, various baselines to define the “normal” temperature are used. In this study, we apply two of the most commonly used baselines. First we apply a fixed baseline, where the climatology is calculated for a 30-year period that is common to all datasets (1993–2022). Second, we use a detrended baseline where the linear trend (calculated based on the full overlap period 1993–2023) was subtracted from the temperature timeseries before performing the MHW detection. The choice of the baseline changes the interpretation of the results and depends on the research question . The fixed baseline maintains the impact of multi-decadal trends, but it is removed using the detrended baseline see for example. How this affects the detected MHWs is discussed in more detail in later sections of this study. We have also tested to account for seasonal variations in the trend, as suggested by , by removing a trend that depends on the day of year. Since the trend (at least in the German Bight) does not show a clear seasonal cycle, the following results are not sensitive to the exact procedure that is used to remove the trend.

To identify the main drivers of MHWs in the German Bight and link them to specific patterns of atmospheric variability we calculate a heat budget for the German Bight region.

The budget is calculated for the uppermost model layer, which is directly related to the SST. In particular in summer, not the entire water column is well mixed during MHWs even in the shallow German Bight. A budget for the entire mixed layer is not advantageous here because VIKING20X does not accurately resolve the mixed layer depth in very shallow regions such as the southern North Sea. This is related to the definition of the mixed layer (e.g. density change relative to a fixed depth) and the relatively coarse vertical resolution. The heat content change associated with a MHW is calculated as the difference in ocean heat content between the minimum within 5 d prior to the MHW onset and the maximum within 5 d after the onset. This ensures that individual MHWs are well separated and that we investigate the weather-related variability that leads to MHWs on timescales of days. Because several local temperature maxima may be reached during one MHW event (that can be caused by different drivers) we focus on the initial onset (first crossing of the temperature threshold) here.

The heat content OHC is calculated from: 1OHC=ρ0cp∫ATdAΔz Here A is the area of the region for which the budget is calculated (e.g. North Sea or German Bight) and T the temperature. Δz is vertical extent of the surface layer and ρ0=1026 kg m⁻³ and cp=3991.87 J kg⁻¹ °C⁻¹ are the reference density and specific heat capacity. The values are taken from the NEMO routine that is used to calculate the surface heat flux. The ocean heat transport is split into a term related to temperature advection (OHTadv) and another related to volume change (OHTvol). This is achieved by introducing a time dependent domain-averaged reference temperature Tref as suggested by and : 2OHT=ρ0cp∫Lu⊥TdLΔz=ρ0cp∫Lu⊥(T-Tref)dAΔz+ρ0cpTref∫Lu⊥dAΔz=OHTadv-OHTvol u⊥ is the velocity normal to a section L. OHTadv is zero if the temperature advected is the same as the domain-average temperature, while OHTvol represents changes in volume transport only. In a closed domain, the divergence of OHTvol (∇⋅OHTvol) is proportional to the horizontal volume transport divergence. In the Boussinesq NEMO model it cancels out with the corresponding vertical term based on the continuity equation. Therefore, the heat budget can be formulated as: 3dOHCdt=∇⋅OHTadv+HFX+VHTadv+Mixing=∇⋅OHTadv+HFX+Res HFX is the surface heat flux and VHTadv the advective part of the vertical heat transport into the surface layer. The surface heat flux is calculated and stored by the NEMO ocean model. This heat content change is then decomposed into the contribution of vertical and horizontal processes. The vertical processes (mixing and VHTadv) are summarized in a single term, named the residual here.

We also calculate the heat budget for the entire German Bight from the surface to the bottom using the same method. In this case Δz is replaced by the integral over all vertical levels. When integrated over the whole water column, vertical redistributions of heat do not change the heat content and the residual term becomes zero (except for a negligible contribution from horizontal diffusion across the domain boundaries).

Relationships between the occurrence of MHWs and the climate indices (and other variables) on seasonal timescales were identified based on conditional probabilities. Note that simply calculating correlations between the indices and the number of MHW days, for example, is not possible, because the number of MHW days is bounded by zero. Even if there would be a strong relationship it cannot be expected to be linear. A strongly negative state of a climate index would not lead to fewer MHW days compared to a neutral state, when the neutral state is already associated with zero MHW days. Instead we calculate the probability of at least one MHW to occur in a season, given that the season mean climate index exceeds its standard deviation (p11). Likewise, we calculate the probability that a MHW occurs, given the climate index does not exceed one standard deviation (p10). The contingency tables are completed with the inverse probabilities (p01 and p00). We then apply Fisher's exact test to the contingency table to test the null hypothesis that the two input variables (occurrence of MHWs and positive state of climate indices) are independent. The null hypothesis is rejected (i.e. the input variables are considered to be related) when the p-value is smaller than 0.05 (5 % significance level).

To identify linear relationships between the season mean temperature anomalies and the climate and pressure indices we apply a linear model defined by: 6y^(t)=a1x1(t)+a2x2(t)+…+anxn(t) y^ is the predicted variable. The coefficients of the linear model (a) are estimated by minimizing the residual sum of squares between the predicted y^ and observed values y using the scikit-learn python package . y is the season mean temperature anomaly in our study. x are the predictor variables, for example a climate or pressure index.

The linear trend is calculated as the slope of the linear regression (Eq.  with one input variable). The p-value is estimated from a two-sided t-test. 7p=2(1-Ft(|ts|;df=n-2)) Ft is the cumulative distribution function of the t-distribution with n-2 degrees of freedom. n is the length of the timeseries. ts is the test statistic given by: 8ts=a1SEa1 with SEa1 being the standard error of the slope a1.

For the Pearson correlation coefficient (r) we calculate the p-value as in Eq. (), but the test statistic is given by: 9ts=rn-21-r2 To account for autocorrelation in the timeseries (i.e. non-independent time steps) when calculating the correlations we replace n by the effective sample size (neff) calculated following .

In both cases, the null hypothesis of the linear trend or correlation coefficient being zero is rejected if the p-value is smaller than 0.05 (5 % significance level).

Although trends in the number of MHW days and intensity are not significant in most datasets when averaged over all grid points in the North Sea, there are regions with significant trends (Fig. a–d). The number of MHWs increases in all datasets (V20, NWS, OIS and OST are shown here as examples) in the Southern Bight (near the entrance to the English Channel) and along the UK east coast. In the model based datasets a significant increase extends into the Dogger Bank region, while in OST significant increases are confined more to the English Channel. OIS shows stronger trends throughout the entire North Sea, but the pattern is similar to the other datasets, with a stronger increase in the Southern Bight and along the UK east coast. Although mostly not significant, an increase is also visible in the German Bight. While in the Southern Bight both, duration and frequency have increased, in the German Bight only the frequency has increased (not shown). Parts of the northern North Sea experienced an increase in frequency, but a decrease in duration, resulting in no change in MHW days.

The maximum intensity shows even more pronounced regional differences. A small part along the French and Dutch coast experienced a significant increase, but most of German Bight and central North Sea show a decrease in the maximum intensity (Fig. e–h). This is interesting as the temperature shows a significant positive trend throughout the entire North Sea, which is largest in the German Bight (Fig. i–l). Although not significant, even in OIS the maximum intensity decreases, despite strong increasing trends in all other variables (including temperature).

The strong change in mean temperature in the German Bight would have led to many more MHW days and more extreme temperature anomalies (i.e. an increase in maximum intensity), had the distribution simply shifted to higher values. However, this is not visible in the characteristics of MHWs (see previous sections). In particular in the German Bight, where the temperature trend is strongest, the number of MHW days did not become significantly larger and the intensity even decreased. Indeed all datasets suggest that the distribution of temperature anomalies has not just shifted, but also changed in width (standard deviation). We have only selected two datasets for an in-depth analysis here to keep the results concise (Fig. ). We use the OST satellite product, because it is commonly used to study MHWs in the North Sea and V20, because it provides the most comprehensive dataset available to us (including for example 3D ocean currents and temperatures as well as all heat flux components).

In order to study temporal changes in the distribution we calculate the mean and standard deviation of the distribution of temperature anomalies in the German Bight within 5-year windows (Fig. ). Anomalies are defined relative to the fixed and detrended baselines as for the MHW detection. We focus on MHWs detected in the German Bight here, as changes in the shape of the temperature distribution were most pronounced for this region. The MHW statistics for the German Bight are derived from the region's mean temperature, as described in the methods section.

Relative to the fixed baseline, the German Bight became about 2 °C warmer than in the early 1980s in all seasons (Fig. a–e). As expected, a higher mean leads to more extremes (MHWs) by shifting the temperature distribution towards higher values.

The North Sea did not just experience a long-term warming trend, but the temperature also varies relative to the trend. This is seen with the fixed baseline, but is better visible when considering temperature anomalies relative to the linear trend (detrended baseline; Fig. f–j). While the years 1989–1993 and 1999–2008 were anomalously warm, the years 1984–1988 and 2009–2013 were anomalously cold.

In contrast to the mean, the standard deviation does not show a significant trend with either baseline, but is dominated by variability on shorter timescales. A strong peak in standard deviation occurred in 1994–1998. Especially in winter and spring the mean temperature was below average relative to both baselines, but still a considerable amount of MHW days occurred (Fig. b, c, g, h). In other years, a lower standard deviation led to fewer extremes than expected from the mean temperature anomaly. For example the period 1999–2003 shows a similar mean temperature anomaly as 2004–2008, but fewer MHW days (Fig. a). This change was particularly pronounced in spring (Fig. c). In winter, changes in the mean were much more important than changes in standard deviation (Fig. b). Anomalies in the mean and standard deviation are not strongly dependent on the season, but their relative importance may differ across seasons.

The characteristics of MHWs show a pronounced seasonal cycle in most datasets used in this study. Most MHW days occur in summer and fall, and fewer MHW days in winter and spring (Fig. a). The seasonal cycle in MHW days exhibits a larger amplitude in the satellite datasets compared to the reanalysis datasets and especially compared to the VIKING20X model. This is caused by too long MHWs in winter and too few MHWs in summer (not shown). The maximum intensity of MHWs peaks in spring and summer, with MHWs being up to 1.2 °C more intense compared to fall and winter (Fig. b). VIKING20X fails to simulate this increase in spring/summer intensity, but shows almost the same intensity in all seasons. The reason for this is related to the vertical structure of MHWs and investigated in detail later. An interesting feature of the seasonal timeseries (here V20 and OST are shown as examples) is that the recent years (2019–2023) show the highest temperature anomaly relative to the fixed baseline, but not the most MHW days (Fig. a). This is caused by a negative anomaly in standard deviation. Additionally, the temperature was below average relative to the linear trend (Fig. f), leading to almost no MHWs relative to the detrended baseline. While the standard deviation was smaller in all seasons, the mean was below average only in spring and summer (Fig. g–j). Both resulted in fewer extremes than expected from the linear long-term warming trend in the German Bight. The reduced standard deviation can also explain why the maximum intensity of MHWs showed a decreasing tendency, even with the fixed baseline. Despite a higher mean temperature, a lower standard deviation reduces the magnitude of the highest temperature anomalies reached.

All results are equally valid for both the V20 and OST datasets. The temporal evolution of the mean is similar between the two datasets, and the same holds for the standard deviation. The number of MHW days can slightly differ within the seasons, but usually show the same change between consecutive 5-year periods (Fig. b–e, g–j). The number of MHW days is very similar when the entire year is considered (Fig. a, f).

MHWs in the German Bight can be driven by either a convergence of the oceanic heat transport, or by a heat gain through air-sea heat fluxes. The latter can be caused by any of the individual heat flux components (latent: LA, sensible: SE, shortwave: SW or longwave: LW), or a combination. In most months the surface heat flux contribution dominates the development of MHWs (Fig. a, d). Nevertheless, the oceanic heat transport does contribute to events as well. For fixed baseline events, the OHT contributes in October, February, and March (up to 75 %). With the detrended baseline, it contributes most strongly in November, December and February (up to 55 %). From May to August the OHT contribution is close to zero or slightly negative for both baselines, meaning it damps the development of MHWs.

For both baselines the SW flux is the dominant driver of MHWs from May to June (Fig. b, e). Additionally, with the detrended baseline the SW flux is the dominant driver of MHWs in August. Most MHWs are either driven by SW alone, or a combination of SW and LA. From August to October, the main driver of most events is an anomalous latent heat flux into the ocean (less evaporative cooling). In winter and spring the results are more baseline-dependent. The ocean heat transport is important with the detrended baseline in November and December, while with the fixed baseline all events in these months are predominantly driven by a combination of the sensible and latent heat fluxes. In February and March MHWs are exclusively driven by the OHT with the fixed baseline, while latent heat flux and sensible heat fluxes play a more important role with the detrended baseline.

For the fixed baseline, MHWs driven by a specific process are scattered across the timeseries. The same is true for SW-driven MHWs with the detrended baseline (Fig. c, f). In contrast, OHT and latent heat driven MHWs tend to cluster for the detrended baseline (Fig. f). Most OHT driven MHWs occurred between 1987 and 1995 and most latent heat driven MHWs between 1998 and 2006 (and additionally multiple events in 2014).

The different drivers of MHWs (here fixed baseline) are associated with distinct atmospheric circulation patterns. We focus on the most frequent types of events here (OHT, SW + LA and LA). Due to their similarity (not shown), SW and SW + LA driven MHWs were grouped together. In addition to the surface conditions, we also investigate the vertical profiles of temperature anomalies during the MHW events.

OHT-driven MHWs (five events) that occur in fall and winter are linked to a low pressure system located northwest of the British Isle (Fig. a). The north-south pressure gradient is increased over the English Channel, driving strong wind stress anomalies that increase the inflow of warm water from the Atlantic Ocean into the southern North Sea. The warm water spreads along the southern North Sea coast and causes strong temperature anomalies in the German Bight (Fig. a, b). Transport-driven MHWs show an almost flat vertical profile with little variations in the temperature anomaly from the surface to 40 m depth (Fig. a–c). Thus for the majority of the German Bight region, the temperature anomaly is similar at the surface and at the bottom. This is expected since these MHWs are driven by barotropic transport anomalies and accompanied by strong winds that cause vertical mixing (Fig. a, b). V20 shows temperature anomalies very similar to those in the other datasets. Note that three different baseline periods are used here because the CST1 and BSH datasets do not cover the same period as the other datasets. The BSH, CST1, G12 and V20 datasets all show very similar profiles, while NWS and the satellite data suggest weaker anomalies for events after 2016 (Fig. b, c). However, only one OHT-driven MHW occurred after 2016 (Fig. c).

Shortwave (+ latent) heat flux driven MHWs (twelve events) that usually occur in the summer months are associated with a high pressure system located over Scandinavia, which leads to low cloud cover (thus high solar insolation) and weak winds over the German Bight. In contrast to the OHT driven MHWs, strong temperature anomalies are not limited to the southern part, but extend across the entire North Sea (Fig. c, d). Shortwave driven MHWs are characterized by a shallow mixed layer of only a few meters depth (Fig. d–f). The high-resolution BSH model and the two reanalysis products simulate a steep temperature gradient in the top 10 m. Consistent with the atmospheric conditions (Fig. c, d), the incoming shortwave heat flux is distributed over a small volume and therefore leads to high temperature anomalies in the top meters of the water column. Even in the presence of tidal forcing (BSH and NWS) the shallow parts of the German Bight are not fully mixed in the model and a thermocline can develop (Fig. d–f). The surface anomalies are very similar in NWS, BSH, and the satellite observations. G12 slightly underestimates the temperature anomaly. V20 cannot simulate such shallow mixed layers, due to its limited vertical resolution. The uppermost level represents a mean temperature of the upper 6 m and cannot capture variations within this depth range. As a consequence, the heat gained at the surface is distributed over a too large volume, causing the MHW intensity in summer to be underestimated at the surface (see Fig. b). At deeper levels the temperature anomalies in V20 generally match those from the other datasets. An important role of the vertical resolution is supported by a very similar temperature profile in CST1 (Fig. f). CST1 is based on a very different model set-up (e.g. including tides, different surface forcing), but shares a similar vertical resolution compared to V20. Below 20 m depth the temperature anomaly is close to zero relative to the 1993–2023 baseline, suggesting a decoupling of surface and bottom MHWs in deeper parts of the German Bight (mostly in the northern part of the area).

Latent heat driven MHWs (eleven events) that often occur in fall are associated with low pressure west of the British Isles and high pressure over Scandinavia or the Baltic Sea. Similar to SW-driven MHWs, positive temperature anomalies can be seen throughout most of the North Sea (Fig. e, f). Winds are weak and blow from south to southeast. As the air masses circulate counter-clockwise around the low pressure system, they bring moist air from the Atlantic to the North Sea. Together with the low wind speed, this leads to highly saturated air close to the sea surface, which reduces evaporation and thus the latent heat loss from the ocean to the atmosphere. Similar to SW-driven MHWs, LA-driven MHWs show the highest temperature anomalies of 1.5 °C at the surface. Compared to the SW driven MHWs the anomaly decreases more gradually to 0.5 °C in 30 m depth (Fig. g–i). This is again consistent with relatively weak winds and a strong heat flux anomaly at the ocean's surface (in this case due to reduced evaporative cooling). Note that depths beyond 30 m are only reached in the northern part of the German Bight region as it is defined here. Again, the BSH model shows a slightly stronger, V20 and CST1 a weaker, vertical temperature gradient compared to the reanalysis products.

These results show that the occurrence of MHWs is linked to specific weather patterns. The main drivers of MHWs depend on the season (Fig. b, e) and so do the corresponding weather patterns (Fig. ). Both the timeseries (Fig. c) and the composites (Fig. ) suggest that the long-term warming trend in the North Sea did not lead to a considerable change in the main drivers of MHWs over time. In general, the identified atmospheric patterns do not depend on the baseline. The same weather patterns drive MHWs detected with the detrended baseline (Fig. ), even though these MHWs appear at different times (Fig. c, f). This suggests that the occurrence of a specific weather pattern alone is not sufficient to trigger a MHW, but variability on longer timescales is important as well. This will be investigated in more detail in a later section.

As the conditions for summer MHWs have already been studied , we focus on the advective-driven MHWs (fall and winter) here. These heatwaves are associated with a low pressure system north of the British Isles (Fig. a, b), but neither the low pressure system nor the heat transport anomalies into the German Bight need to be exceptionally strong to trigger a MHW. This is illustrated by three examples (Fig. ).

In 1990 a low pressure system over Scotland caused strong wind anomalies over the English Channel (Fig. a), which resulted in a strong heat transport anomaly (Fig. : black lines). However, the negative surface heat flux anomaly (ocean heat loss) damps the transport anomaly throughout the entire southern North Sea, preventing a MHW (Fig. b).

Another mechanism is seen in 2009. The wind speed (Fig. e) and inflow (Fig. : blue lines) anomalies in the English Channel are even stronger and more persistent than in 1990. The heat flux anomaly in the German Bight is positive and therefore also contributes to a warming (Fig. d). Still, no MHW is detected in the German Bight. This is caused by an initially cold ocean (relative to both baselines; Figs. f and b, c). In late summer 2009 the German Bight was about 1.5 °C colder than the climatology. On 14 November, when the shown low pressure system reached the North Sea the temperature anomaly was close to zero relative to the detrended baseline (Fig. f). Thus, the very strong anomalies in 2009 caused a strong increase in the temperature of the German Bight, but did not lead to a MHW (Fig. b, d).

In 1995 a low pressure system northwest of Scotland (upper left boundary of the map) led to an increased pressure gradient and slightly stronger than average southwesterly winds over the English Channel (Fig. g, h). Neither the low pressure system itself, nor the wind anomalies, nor the heat transport anomalies in the southern North Sea were particularly strong or sustained for an unusually long time (Fig. : red lines). Still, a MHW developed from the relatively small heat inflow anomaly (with a minor contribution from the surface heat flux), as the temperature was already close to the MHW threshold before (Figs. i and c). This initial anomaly was caused by a stronger low pressure system that passed the northern North Sea about 20 d prior to the MHW onset (Fig. a: red line around day 280).

This shows that advective MHWs in the German Bight are driven by a rare combination of preconditioning, inflow, and heat flux anomalies rather than extreme atmospheric conditions (e.g. a very strong low pressure system close to the British Isles). The preconditioning factor is strongly influenced by the chosen baseline, while the change in temperature is independent of the baseline (Fig. b, c). With the detrended baseline the temperature was close to the MHW threshold when the second low pressure system passed on day 300, while it was just above the climatology for the fixed baseline. The same warming then raised the temperature above the MHW threshold for the detrended baseline, but barely above the climatology with the fixed baseline.

As the number of MHW days is bounded by zero, a linear relation (i.e. correlation) to the other timeseries cannot be expected. Instead, we calculate the conditional probabilities of at least one MHW occuring in a season, dependent on a given state of other variables. We group the months by the dominant processes that cause MHWs (Fig. ), as we expect similar atmospheric patterns to be associated with a warming/cooling of the German Bight. These are winter (JFM), summer (MJJ), early fall (ASO) and late fall (ND). The month of April is not contained by any group, because it is a transition month and does not fit the winter nor the summer group.

In all seasons the occurrence of MHWs is strongly related to the season mean temperature anomaly, even though MHWs themselves can be as short as 5 d and do not necessarily cause an above average season mean temperature (Fig. a).

Although the surface heat flux is significantly correlated to the NAO in most months, except early fall (Fig. ), an anomalously positive NAO index is not related to a higher probability of MHWs occurrence (Fig. b). In summer there is a significant negative correlation to the heat transport convergence, because a local heating of the German Bight by surface heat fluxes increases the temperature of the outflow across the northern boundary (Fig. ).

The EAP is significantly correlated to the heat transport convergence in all seasons except summer (Fig. ). Thus, in contrast to the NAO, the EAP influences the heat budget by changing the (wind-driven) oceanic circulation rather than the atmospheric conditions alone. For both heat transport convergence and heat content change correlations to the EAP are similar year round, with values of 0.4–0.5 for the convergence and 0.2–0.4 for the net heat flux (p-values are close to 0.05). In late fall and winter the EAP strongly controls the heat transport along the southern boundary of the North Sea from the English Channel to the western boundary of the German Bight. In late summer the EAP index is not significantly correlated to the heat transport across the western, but the northern, boundary of the German Bight (Fig. ). Still, only in winter MHWs are significantly more likely to occur given that the EAP is in a positive state (Fig. c). The same is true for the probabilities of MHWs given the NAO and EAP are anomalously positive at the same time (not shown).

The ENSO index is significantly correlated to the heat transport convergence in late fall (Fig. ), but no relation to the probability of MHWs is found (not shown). There is no significant correlation to the AMV. Because the AMV is based on 10-year low-pass filtered temperature timeseries, it is not expected to explain variability on seasonal timescale (Fig. ).

Therefore, especially the NAO and EAP are linked to heat content changes, but their state does not significantly change the chances of MHW occurrence in most seasons. As argued above for individual MHW events, the initial temperature is also important on seasonal timescales. The initial temperature is defined here as the mean temperature anomaly of the month prior to the season to avoid a large impact of occasionally occurring MHWs that extend across seasons. In most seasons the probability for MHWs is significantly higher, if the month prior to the season was anomalously warm (Fig. d). In summer MHWs occur slightly more often without an anomalously warm initial temperature, which causes the p-value to be just above the significance threshold (p-value = 0.06). As will be discussed in more detail below, season mean temperature and initial temperature are closely related, especially in MJJ. Still, there are years in which the season mean temperature anomaly is high, but the initial temperature is neutral or low. The significant dependency for the season mean temperature (Fig. a), but non-significant dependency for the initial temperature (Fig. d) in MJJ reflects that in these particular years MHWs have occurred (applies to 4 years). This is reasonable because years that had a below average initial temperature anomaly, but still a higher season mean temperature anomaly were likely years with exceptionally strong surface heat flux forcing that also triggered MHWs.

Since the season mean temperature anomaly strongly influences the occurrence of MHWs, we now investigate what drives anomalously warm seasons.

In all seasons except winter, the initial conditions show a higher correlation with the season mean temperature anomaly than the prevailing weather patterns (pressure and climate indices). The correlation is significant with values exceeding 0.43 (p ≪ 0.05). Especially in summer, the initial temperature has a strong impact and the correlation exceeds 0.8 (Fig. e–h).

Linear combinations of the EAP and NAO (and ENSO for late fall) yield a significant correlation to the season mean temperature anomaly only in winter and late fall (Fig. e–h). Instead of using climate indices, it is also possible to define a pressure index that directly reflects the local atmospheric conditions over the North Sea. The pressure indices were defined to maximize their correlation to the heat content change during the respective seasons. Only in summer this leads to a higher correlation to the season mean temperature compared to the EAP and NAO (Fig. e–h).

When the climate or pressure indices are linearly combined with the initial temperature anomaly, correlations exceed 0.66 (p ≪ 0.05) in all seasons, except in early fall. The corresponding timeseries are shown in Fig. a–d. In early fall the correlation is lower (0.61; p ≪ 0.05) because neither the established climate indices nor the local pressure index are significantly correlated to the season mean temperature anomaly (Fig. g). This suggests that a warm fall can be driven by various weather patterns. Either by shortwave radiation (as in summer), latent heat fluxes, or oceanic heat transport (as in late fall). This is not well captured by the season mean climate or pressure indices. Removing the month of August or October from the group does not improve the performance of the linear model (not shown), which further suggests that the transition between the dominant driver of heat content changes can vary from year to year.

The dependency of the seasonal temperature anomalies and occurrence of MHWs on the initial temperature gives rise to variability on longer than seasonal timescales, even without low-frequency variability in the forcing itself and very different mechanisms that govern the temperature evolution in different seasons. Four example periods are selected to illustrate this. We analyze the heat budget for the entire water column here, such that vertical redistribution of heat does not play a role (see methods section for details on the heat budget calculation).

In 1987, the German Bight is in a cold state until it strongly gains heat in late fall by a heat transport convergence, which is followed by a strong surface heat flux anomaly (Fig. a). The heat content anomaly reaches a peak in winter 1987/88 resulting in a long-lasting MHW. A large amount of heat is lost again in spring 1988, until the atmosphere drives another strong increase in ocean heat content and multiple MHWs at the beginning of 1989.

In spring 1996, the OHC is strongly negative in the German Bight (Fig. b). Until 1998, the heat content gradually increases due to positive surface heat flux anomalies in summer and positive heat transport anomalies in winter. In fall 1997, two MHWs are triggered and several more MHWs occur after a further increase in OHC in 1998.

In winter 2010/11 a strong minimum was reached due to a strongly negative surface heat flux and heat transport anomalies. The recovery from the minimum was initiated in January 2011 by anomalous heat gain from the atmosphere (Fig. c). The heat content change was large, but due to the cold initial state it did not lead to any MHWs. Afterwards the heat content remained stable due to offsetting heat transport and surface flux anomalies. After reaching another local minimum, a strong heat gain set in during May 2013. The increase in OHC was first driven by the atmosphere, before anomalous oceanic heat convergence took over. A maximum in heat content that coincided with multiple MHW events was reached in spring 2014.

The most recent years (2019–2023) are characterized by little variability in OHC (Fig. d), consistent with the reduced standard deviation of temperature anomalies shown before (Fig. ). In particular, the surface heat flux anomalies are small. Ocean heat transport anomalies are similar in magnitude compared to other periods, but often compensated by negative surface heat flux anomalies. Only in fall 2022 and 2023 does a strong ocean heat transport anomaly, together with a weakly positive surface heat flux anomaly, increase the ocean heat content and lead to short MHWs. Surface heat flux and heat transport anomalies frequently change sign, such that there is no gradual build-up of heat over multiple seasons either. Instead, the OHC fluctuates around the mean (anomaly of zero) on shorter timescales.

The examples clearly illustrate how the history of surface heat flux and heat transport anomalies influence the heat content of the German Bight and contributes to preconditioning for MHWs over multiple seasons. A peak in ocean heat content on interannual to pentadal timescales may be reached by a strong heating anomaly during a single season (e.g. winter 1987/88), that is maintained in the following seasons. In other periods, however, a peak in OHC results from a more gradual build up of heat anomalies over multiple seasons within a year (e.g. May 2013 to April 2014) or over multiple years (e.g. from March 1996 to July 1998). Often MHWs follow periods of sustained positive surface heat flux anomalies in summer and positive heat transport anomalies in the following winter (e.g. 1996–1998 or 2013–2014). Therefore, it is usually not the conditions (e.g. NAO/EAP) in a single season that matter for the occurrence of MHWs, but the prevailing (and cumulating) atmospheric conditions over multiple seasons.

The North Sea experienced a strong warming in the late 1980s and has continued to warm (at a lower rate) afterwards . This temporal evolution is well represented by a number of satellite, reanalysis and model-based datasets, such as VIKING20X. Along with the long-term warming, MHWs became more frequent in the North Sea relative to a fixed baseline, consistent with previous studies . Due to large interannual variability, only trends in frequency are significant in most, but not all, datasets when the overlap period of all datasets is considered (1993–2023). The duration and number of MHW days have increased, but the linear trends are not statistically significant in most datasets. When the full satellite period (1982–2023) is considered in datasets that cover this period, trends in frequency, duration, and MHW days are significantly different from zero for most of them. In contrast, the maximum intensity has not increased. For both the overlap and the full satellite period trends in maximum intensity are not significantly different from zero (and tend to be negative) in any dataset. While interannual variability shows very high agreement between different satellite and model-based datasets with correlations exceeding 0.8, linear trends can differ. Such a result was also obtained for the Indo-Pacific region . Especially the OIS dataset shows stronger trends in the temperature and MHW characteristics than other datasets in the North Sea. This is an important result, because the OIS dataset is used for example by as the only dataset to assess past trends in North Sea MHWs. Although we don't expect their results to fundamentally change with other datasets, their trend estimates are likely at the upper limit. As expected, the variability in the reanalysis datasets, that both assimilate the OST surface temperatures in this case, is very similar. Still, trends show slightly different patterns and lower values, suggesting that the model choices play a role too. Simulations with the forced ocean model VIKING20X that does not assimilate ocean observations shows a very similar temporal evolution (variability and trends).

In agreement with , the increase in MHW frequency is largely caused by the linear temperature trend. Relative to a detrended baseline we find no linear increase in the MHW characteristics in any of the datasets. Especially the years after 2019 showed only few MHWs. This suggests that natural sub-decadal variability has damped the effects of the long-term trend in recent years.

In line with previous studies, MHWs in late spring and summer are usually driven by increased shortwave radiation and reduced vertical mixing in the ocean, associated with a high pressure system over Scandinavia . A somewhat similar weather pattern can lead to MHWs driven by anomalous latent heat flux that typically occurs in late summer/early fall. Most winter MHWs are driven by the ocean heat transport and associated with a very different weather pattern. A low-pressure system located northeast of Scotland and strong south-westerly winds over the English Channel drive a large amount of warm Atlantic water into the southern North Sea.

Consistent with the drivers of MHWs, they have different vertical structure. Shortwave driven MHWs occur in the top few meters of the ocean with much smaller anomalies below. This is consistent with a case study by and a decoupling of SST anomalies and vertically integrated OHC in summer . argue that MHWs themselves are an important driver of stratification in the shallow southern North Sea that is usually well mixed by tides and wind/wave induced mixing. This has important consequences, since many mobile marine species may be able to avoid the high surface temperatures and benthic species are only affected in very shallow regions close to the coast. For the German Bight, these regions are mostly tidal flats and experience much larger variability caused by a regular exposure to the air temperature. Due to the strong temperature gradient near the surface, models with coarse vertical resolution and no data assimilation (V20 and CST1) underestimate the intensity of MHWs. They simulate a mean temperature of the surface layer (here 5–6 m thick), which is not representative for the temperature in the top meter during summer MHWs. In contrast, the BSH model, which does not use data assimilation either, shows maximum temperature anomalies much more comparable to the satellite-based datasets. The presence (CST1 & BSH) or absence (V20) of tidal mixing seems to be less important than the vertical resolution (and potentially vertical mixing parameterizations). Similarly, latent heat driven MHWs show stronger anomalies at the surface, but the vertical temperature gradient is smaller compared to SW-driven MHWs. MHWs in winter are often driven by ocean heat transport anomalies under strong winds and therefore the water column is well mixed in the German Bight. There are no substantial biases between the datasets for these MHWs, resulting in an almost flat seasonal cycle of MHW intensity in V20, whereas satellite measurements show pronounced seasonal cycles.

While MHWs are associated with distinct atmospheric patterns they are not a sufficient condition for MHWs to occur. Even exceptionally strong atmospheric anomalies that are associated with strong heat flux (i.e. heat content change) anomalies do not necessarily result in MHWs. The reason for this result is that the temperature and equivalently the ocean heat content are integrated properties. A given (surface or lateral) heat flux changes the temperature, but whether the temperature is anomalously high (i.e. a MHW occurs) depends on the history of the fluxes. Weather related variability on timescales of days to weeks is strongly linked to the fluxes (e.g. solar insolation, stronger heat transport), that is changes in temperature, but not absolute temperature anomalies. Therefore, processes that lead to a warming of the ocean on longer, seasonal to decadal, timescales (ocean preconditioning) play a major role for the occurrence of MHWs.

With the detrended baseline, this ocean preconditioning is related to seasonal to decadal variability. With the fixed baseline, the long-term warming trend acts as an additional preconditioning factor. Which timescales contribute to the trend versus variability relative to the trend depends on the chosen period for detrending and analysis. In this study, the period is 1993 to 2022 (30 years) and thus multidecadal variability (e.g. the AMV) contributes mostly to the trend, while variability on interannual timescales (e.g. NAO) mostly contributes to variability relative to the trend. Therefore, the long-term trend itself probably contains anthropogenic forcing, but also low-frequency natural variability. The AMV index has increased from the mid 90s up to now, which is exactly the same time period of the linear trend in our and most other MHWs studies in the North Sea e.g.. It is also interesting to note that temperature is often assumed to show larger variability on longer timescales red spectrum;. Therefore, the longest timescale covered by a timeseries is expected to dominate the occurrence of MHWs.

On seasonal timescales the occurrence of MHWs is linked to the prevailing weather conditions during the season, but more importantly to the initial temperature (i.e. the temperature evolution in the prior season). Especially the summer mean temperature anomaly is almost entirely explained by the temperature in late winter/ spring. This is consistent with and , who found the summer temperature to be highly correlated with the atmospheric conditions and SST in previous seasons. A reason for this observation is a strong anti-correlation between different heat flux components and with the ocean heat transport. A summer with strong solar insolation is often accompanied by more heat loss through the transport and/or the other surface heat flux components. A similar damping mechanism was already observed by in an idealised 2-layer model of the North Sea.

Consistent with the view that temperature is an integrated quantity (i.e. it is set by temporally integrated fluxes) and climate indices being more related to temperature changes than absolute temperature anomalies, we only find a statistically significant dependency of MHWs on the state of the EAP in winter. For other climate indices previously found to be important , such a dependency was not found. For the AMV this is likely explained by the fact that used a fixed baseline. As mentioned above the AMV showed an almost linear increase from the mid 1990s to 2020s and therefore aligns with the temperature trend in the North Sea. Here we used a detrended baseline to explain the natural variability contribution on interannual to decadal timescales. In agreement with we find the NAO to be strongly related to the surface heat flux in winter and the EAP to the inflow of warm water through the English Channel. The NAO is not related to the heat transport in the southern North Sea in most seasons (including winter), which was also noted by and . Thus a link between NAO/EAP and MHWs exists, but the NAO and EAP being positive is not a sufficient condition for MHWs to occur. The temperature anomaly at the beginning of the seasons is more important in most seasons, except for the winter months (JFM).

The strong impact of the season's initial temperature gives rise to the strong impact of low-frequency variability. The North Sea can build up heat over several seasons or even several years. More detailed analysis based on the model output shows that the surface heat flux and oceanic heat convergence both contribute to heat content changes. As most of the climate indices do not have the same influence year round, a positive NAO winter is associated with a warming North Sea in winter, which is not necessarily the case for a positive NAO year. In summer for example, a positive NAO state is correlated to a stronger surface heat flux, but the effect is compensated by a negative correlation to the oceanic heat transport convergence. Therefore, also on longer timescales a single climate index can not explain the evolution of the German Bight's temperature. It is rather a series of positive NAO/EAP winters that leads to a warm state of the southern North Sea.

The inflow of warm water through the English Channel has a strong influence on the southern North Sea and MHWs in the German Bight, consistent with and . Especially variations in transport are important, but variations in the inflow temperature also play a role. Still, our sensitivity experiments suggest that SST in the North Sea (especially in the German Bight) is almost fully determined by the atmospheric state on decadal and shorter timescales, consistent with results obtained by and . The reason for this result is that the transport is largely driven by the local winds over the English Channel and the temperature of the inflow is set by the local surface heat flux . Even strong temperature anomalies in the eastern Atlantic do not enter the North Sea under a prescribed atmospheric state, because they are quickly damped by the surface heat flux.

Nevertheless, our forced ocean model experiments are not able to answer the question how remote changes that are for example related to the large-scale Atlantic circulation feed back on the local atmospheric state in the North Sea. In reality a changing surface temperature in the eastern Atlantic affects the surface heat fluxes that, in contrast to our simulations, change the atmospheric conditions. Therefore, it cannot be concluded based our experiments that the temperature outside the North Sea is not important. As noted above, find a similarly strong dependency of the German Bight's temperature on the local atmospheric conditions, but they use a forced ocean model as well. Whether variability of the Atlantic Ocean circulation, for example, is linked to temperature variability in the North Sea therefore remains to be studied in a coupled model set-up.

This result has several implications. Forced simulations are very useful if the goal is to understand the past evolution of MHWs with physically consistent ocean and atmosphere states. That the temperature evolution is strongly determined by the atmospheric state is very useful for regional models, because not simulating interannual variability outside the North Sea does not introduce errors in the central and southern North Sea. The strong dependency on the atmospheric state is consistent with all model datasets showing a very similar evolution of MHWs, despite very different set-ups. Furthermore, atmospheric observations are readily available and contain a lot of information about the ocean's temperature. With an accurate initial ocean state, predictions of the pressure field would allow for potential predictability of MHWs. Given that have shown that skillful predictions of local atmospheric variables are possible, this could also allow the prediction of MHW likelihood. Note that on seasonal timescales the initial state already contains a lot of information on the likelihood of MHWs.

Nevertheless, when the goal is to understand the full physical mechanisms involved in the evolution of MHWs, i.e. the initial source of temperature anomalies, coupled model experiments are required. The large-scale oceanic circulation could play an important role in shaping the atmospheric state over the North Sea, such that the evolution of MHWs could be strongly, but indirectly, influenced by remote oceanic variability.