At the heart of the dynamical slab ocean model lies the conservation of energy. The temporal evolution of the surface layer temperature (∂Ts/∂t) at a particular oceanic grid point is governed by the net heat fluxes, expressed as: 1ρCpH∂Ts∂t=Fsurf+Fo+Qflux, where ρ is the density of seawater, Cp its specific heat capacity (refer to Table  for associated values) and H the depth of the mixed layer. The left-hand side of Eq. () represents the rate of change of heat content per unit area of the slab ocean. The right-hand side includes the net surface flux Fsurf computed by the atmosphere model, an ocean heat flux term Fo (horizontal and vertical transport) computed by our dynamical slab ocean model and an (optional) additional forcing (Qflux) that accounts for other processes. The net surface flux is given by Fsurf=Frad+Fsens+Flat, where Frad includes net shortwave and longwave radiation, Fsens represents enthalpy exchange due to temperature differences between the ocean surface and the air, and Flat captures latent heat exchange during phase changes of water. The Qflux term can be used in the absence of Fo to approximate the effects of ocean mixing and dynamics, but it can also represent other processes depending on the simulated planet (e.g., geothermal heating, important in exoplanet contexts). A positive (negative) sum of Fsurf+Fo+Qflux leads to a net gain (loss) of heat, increasing (decreasing) the ocean surface temperature.

Sea ice develops when the ocean temperature falls below the freezing point of seawater (Tfreeze=-1.8 °C)

This value reflects modern Earth's average ocean salinity of 35 psu. The model's Tfreeze value can be changed as per the seawater salinity of the model planet.

The evolution of ice extent and thickness is governed by energy conservation during phase changes, ensuring that the ocean temperature remains at Tfreeze as long as ice is present. Specifically, if the temperature of the surface ocean layer (Ts) instantaneously falls below its freezing point, its temperature is set back to Tfreeze and the resulting energy difference creates ice. Conversely, if the ocean temperature exceeds Tfreeze when ice is present, the temperature is set back to Tfreeze, and the excess energy is used to melt part of the ice. A similar mechanism governs melting from above. If snow has accumulated on top of the sea ice and the surface temperature (of snow-covered ice) exceeds Tmelt=0°C, it is reset to Tmelt and the energy difference is first used to melt the snow and then the ice. If the heat fluxes are strong enough such that energy is still remaining, it is used to warm the ocean, thus satisfying energy conservation. Sea ice may grow or melt primarily by adjusting either its area or thickness, depending on whether the dominant energy flux is from the ocean below or the atmosphere above.

As described above, snow and ice are both subject to surface heat fluxes from the atmosphere (Fa-i) and conductive fluxes from the ocean and within the ice itself (Fi-o). Ocean heat transport or q-flux changes the ocean temperature, and so, only acts indirectly on the ice. Following , we rewrite Eq. () to specifically describe the evolution of the ice temperature Ti: 2ρiCiH∂Ti∂t=2[Fa-i-Fi-o] where ρi is the ice density, Ci is its specific heat capacity, and Fi-o=λH(Ti-T0) is a vertical conductive flux, where λ is the thermal conductivity of ice (refer to Table  for associated values). The factor of 2 accounts for the linear (conductive) thermal profile within the ice. If a snow layer is present, we also account for the associated snow-ice conductive flux.

The parameterisation of sea ice and snow albedo also plays a critical role in shaping a model planet's climate and consequently, its observability. addressed this by treating the albedo of bare sea ice as a function of ice thickness. In their framework, the surface albedo (A) at an oceanic grid point could correspond to that of open ocean, overlying snow (if present), or bare sea ice, the latter varying with thickness according to: 3A=Aicemax-Aicemax-Aicemine-hice/hice0 In the present study, we improve this scheme by incorporating a spectral dependence of albedo in addition to the existing thickness dependence. Figure a shows how the model's prescribed sea ice albedo varies as a function of ice thickness in the Visible (VIS; 250–690 nm; green line) and Near-Infrared (NIR; 690–4000 nm; red line) spectral bands. The spectrally-independent albedo profile from is also shown for reference (dashed green). This earlier scheme assigned an albedo of 0.2 to newly formed ice, whereas our formulation introduces a physically motivated transition, with the albedo increasing smoothly from its minimum possible value, i.e., Aicemin≡Aocean=0.07,

The albedo of open water primarily results from Fresnel reflection and is largely wavelength-independent, consistent with measurements e.g.,.

The nomenclature “dynamical slab ocean model” is purposefully chosen to reflect our model's emphasis on ocean dynamics. Figure  provides a cross-sectional view of the ocean model in the horizontal (latitude) and vertical (depth) dimensions, and illustrates the model's various OHT regimes. The upper section depicts the oceanic mixed (surface) layer, with a fixed thickness of Hs=50 m, where interactions with the atmosphere occur (see Sect.  for a discussion of the implications of using a fixed mixed-layer depth). The lower section corresponds to the deep ocean, with a thickness of Hd=150 m, which exchanges heat only with the surface layer.

In our model, heat transport by the ocean is represented by four components, detailed through Sect. –:

Figure a shows the decadal-averaged ocean heat flux from the OHT-on simulation, with net positive values indicating atmospheric heating by the ocean (ocean cooling, upward heat flux) and negative implying atmospheric cooling by the ocean (ocean warming). Alternatively, this plot can be seen as the vertically-integrated convergence of the OHT. We see that our model correctly produces surface ocean cooling (or OHT convergence) in subtropical and high-latitude oceans, and warming in the equatorial region. However, the particularly strong regions of cooling over the Gulf Stream and the Kuroshio currents that are observed on Earth are not captured, since the return flow of the western boundary currents is not explicitly included in our model.

Figure b displays the decadal-averaged SST difference between the OHT-on and OHT-off simulations, effectively capturing the thermal response of the surface ocean. There is a clear spatial correspondence between the ocean heat flux (the forcing) and the SST (the response), particularly in the tropics, highlighting the role of OHT in shaping surface temperatures. The colour map is centred around ΔT=0 °C to emphasise regional warming and cooling. One of the most prominent features of Fig. b is that equatorial SSTs are 5–8 °C cooler in the OHT-on simulation. As in the aquaplanet case (Fig. ), this pronounced cold tongue – especially visible in the eastern Pacific – is primarily driven by Ekman-induced upwelling. A similar structure is also evident in observations for e.g.,. Our more eastward cold tongue compared to is likely a consequence of Sverdrup dynamics. used a purely frictional Ekman balance at the equator, and noted that the position of the cold tongue is sensitive to the frictional damping coefficient ϵ (see Sect. ), with lower values of ϵ favouring an equatorially centered, zonally extended cold tongue. In our study, the introduction of a Sverdrup balance near the equator suppresses unrealistically strong cross-equatorial frictional transport (see Sect.  for more details), shifting the equatorial circulation into a curl-driven regime and anchoring the upwelling more strongly to the eastern basin. Meanwhile, mid-latitude SSTs are warmer when OHT is enabled, primarily due to horizontal diffusion, with some contribution from GM advection (also seen in Fig. ). The Southern Ocean warms by 7–10 °C in the OHT-on case, consistent with and , again driven by GM advection and horizontal diffusion. Our OHT-on simulation also yields warmer and more realistic tropical SSTs than those in , with temperatures comparable to NCEP/NCAR reanalysis.

While OHT acts as the forcing and SSTs the response, the downstream consequence is seen in precipitation patterns. Figure c shows the corresponding decadal-averaged precipitation difference. As in the aquaplanet simulations, the OHT-off case features a precipitation peak at the equator. However, when OHT is enabled, we get a (weak) double-peaked precipitation structure around the equator due to suppressed evaporation given the Ekman-driven upwelling. Following , we note that lower values of the frictional damping coefficient ϵ (see Sect. ) lead to a more zonally extended equatorial cold tongue.

The annually-averaged position of the ITCZ on Earth lies near 5° N . This hemispheric preference arises due to a combination of factors, including the unequal distribution of landmasses and the oceanic Meridional Overturning Circulation (MOC). The latter plays a particularly important role: demonstrated that the ITCZ shifts northward in GCM simulations when OHT is included, even in the absence of continents. This suggests that the northward displacement of the tropical rain band is not merely a response to hemispheric radiative asymmetries, but is actively maintained by the MOC, which transports approximately 0.4 PW of heat across the equator into the Northern Hemisphere, thereby shifting the ITCZ northward.

In our OHT-off simulation, the absence of OHT results in a near-equatorial ITCZ, slightly north-shifted (≈ 3.5° N). This offset is likely a model artefact. Figure c shows the precipitation difference between the OHT-on and OHT-off simulations: a negative anomaly in the Northern Hemisphere and a positive one in the Southern Hemisphere indicate a southward migration of the ITCZ when OHT is enabled. In the OHT-on case, the ITCZ is positioned around 3.5° S. This shift arises from: (a) the lack of density-driven MOC in our model, and (b), enhanced Southern Hemisphere warming due to OHT-driven sea ice loss. This Southern Hemisphere warming drives a northward atmospheric energy transport, which shifts the ITCZ to around 3.5° S, as shown in Fig. c. This behaviour is consistent with the compensation mechanism discussed in , where the ITCZ migrates toward the hemisphere receiving more net heating.

Figure  presents the meridional ocean heat transport (in petawatts) for modern Earth, comparing results from our model (solid red line) with estimates derived from ECMWF reanalysis (black dashed line; ). While our model does not capture the full magnitude of Earth's heat transport, it successfully reproduces the wind-driven component of the circulation well due to the combination of the Ekman and GM parameterisations. This is especially evident in the Northern Hemisphere tropics, where the Pacific Ocean dominates the total transport (see Fig. 5 in ).

However, the peak northward OHT in our model (≈ 1 PW) is lower than reanalysis estimates (≈ 1.5 PW), and Southern Hemisphere OHT is also underestimated (-0.3 PW vs. -1.5 PW). These discrepancies stem primarily from the absence of a full-depth MOC in our model. While we include Ekman transport, horizontal diffusion, and the GM parameterisation that together represent key aspects of the wind-driven and eddy-driven components of the MOC, we do not simulate density-driven processes such as deep water formation and diapycnal mixing. These components are critical for reproducing the full strength and structure of the observed MOC, particularly in the Atlantic and Southern Ocean, where density-driven transport dominates (see Fig. 5 in , and also and ).

The seasonal evolution of extrapolar SSTs (60° S–60° N, Fig. a) and global sea ice coverage (Fig. b) is strongly modulated by the presence or absence of OHT, which affects both the amplitude and phase of their annual cycles. Figure  compares these diagnostics across three cases: observations (black), the model with OHT enabled (red) and disabled (blue). Shaded regions represent the 2σ interannual variability over a 30–40 year period

The observed SST variability is larger than in the model due to (a) the trend of rising atmospheric CO₂ levels between 1982–2022, and (b) multi-year climate variability, for instance, the El Niño–Southern Oscillation (ENSO).

Observed SSTs (from NOAA, Fig. a) exhibit a characteristic bimodal seasonal pattern, with peaks around the equinoxes, accompanied by corresponding advances and retreats in global sea ice coverage (NSIDC, Fig. b), reflecting seasonal polar heating and cooling. The physical origins of the observed SST peaks are rooted in hemispheric asymmetries. The first peak around the March equinox arises from the warming of the vast Southern Hemisphere oceans, whose influence outweighs that of the colder, but smaller Northern Hemisphere ocean area during that time of the year. The second, smaller peak near the September equinox is due to Northern Hemisphere warming, partially offset by cooler Southern Hemisphere SSTs. Our OHT-on and -off simulations reproduce this bimodality to differing extents, but we produce a strong SST peak around the September equinox. This is because our Southern Hemisphere oceans remain too warm due to underestimated Antarctic sea ice (see Fig. ). Additionally, we tend to lag observations by about a month likely due to a slightly elevated thermal inertia in our model ocean and/or the assumption of a fixed mixed-layer depth. The latter overly damps SST variations in regions where the real ocean mixed layer would be shallower – like in the tropics. We refer the reader to Sects.  and for more details.

When OHT is disabled, the climate becomes 1.5–2 °C cooler than observed (Fig. a), with a muted seasonal SST cycle. Sea ice coverage is unrealistically high year-round – by approximately 10 million km² (Fig. b) – since the absence of poleward heat transport allows ice to grow in both area and depth. This thick sea ice layer suppresses warming and inhibits ocean-atmosphere coupling, damping SST variability further. Conversely, enabling OHT improves latitudinal heat redistribution, leading to warmer extrapolar SSTs and limiting sea ice extent to values much closer to observations. Although the OHT-on simulation does not fully capture the observed bimodality in sea ice, it more accurately captures the seasonal SST amplitude and the annual averaged values of both SST and sea ice coverage. The annual mean biases are approximately 0.6 °C for SST and 3 million km² for sea ice coverage, respectively. The extent of sea ice also influences the planetary bond albedo, which in our OHT- on and -off simulations, is approximately 0.32 and 0.33 respectively, closely matching the value of 0.31 derived from the NCEP/NCAR Reanalysis R-1. These represent a substantial improvement over the value of 0.36 obtained in , who attributed their high albedo value to an excessive amount of clouds produced at the ITCZ. Our lower and more realistic albedo therefore suggests that we have a reduced ITCZ cloud bias, although a detailed cloud radiative analysis is beyond the scope of this work.

Figure b shows that the global median sea ice extent in the OHT-on simulation varies between 16–23 million km² – much closer to the observed range (18–27 million km²) than the OHT-off case. Additionally, in contrast to , we retain Northern Hemisphere sea ice throughout the year, whereas in their setup it disappeared completely in the summer. However, both our OHT-on and -off cases still overestimate Northern Hemisphere sea ice (see Fig. ), consistent with findings from . This likely stems from three factors: (a) the absence of a deep overturning circulation transporting heat from the Southern Ocean to the North Atlantic, contributing to both, excess sea ice in the North, and its lack in the South, (b) biases in sea ice albedo (see Sect. ), and (c) the lack of sea ice drift, which promotes excess ice build-up. In contrast, Antarctic sea ice (Fig. ) is underestimated in both model configurations. The same factors (a) and (b) apply here, as well as the missing Antarctic Circumpolar Current (ACC), which, in the real climate system, helps thermally isolate the Antarctic continent and sustain sea ice cover.

We remind the reader that the primary objective of our model is not to exactly reproduce modern Earth's climate, but to capture its first-order features. This ensures greater confidence in the model's behaviour when applied to different planetary contexts, whether paleoclimate or exoplanets. Nevertheless, our modern Earth simulations yield surface temperatures that compare well with both observational datasets and previous modelling studies (see Sect. ). With OHT enabled, the simulation produces a global annually averaged surface temperature of 13 °C, in close agreement with the NCEP/NCAR reanalysis temperature of 14.0 °C for the 1981–2010 period. In contrast, the simulation without OHT yields a cooler average temperature of 12 °C. , by comparison, obtained a warmer OHT-enabled mean temperature of 14.9 °C. They note that this likely results from a combination of an overly strong meridional heat transport and an enhanced greenhouse effect due to a larger amount of high clouds, as well as the seasonal disappearance of Arctic sea ice in summer. In our work, Arctic sea ice is retained throughout the year, which contributes to a slightly cooler climate.

Beyond global average temperatures, the inclusion of OHT also improves other key diagnostics, such as extrapolar SSTs and total sea ice extent, bringing them into closer alignment with observations. These quantities play a central role in the planet's energy balance–through their influence on albedo and outgoing longwave radiation–and are especially relevant for exoplanet climate modelling. Given the limited constraints on exoplanet climates, such agreement with Earth benchmarks is a promising validation of our model's physical realism.

One of the additions we introduce to the Generic-PCM's dynamical slab ocean model is the Gent-McWilliams (GM) parameterisation . The GM scheme enables isopycnal mixing and modifies the large-scale ocean temperature structure (Fig. ) through horizontal and vertical heat redistribution. In our two-layer model, this redistribution alters the temperature contrast between the surface and deep layers that drives Ekman transport (see Sect. ), thereby introducing a coupling between eddy-induced and wind-driven circulation.

To assess the relative importance of GM and convective adjustment, we compare a hierarchy of aquaplanet simulations that systematically include or exclude these processes. Figure a shows the decadal-averaged zonal deep layer temperatures of the relevant simulations. Case 6 (horizontal diffusion + Ekman + GM; solid black line) exhibits substantially cooler deep layer temperatures at mid- to high latitudes than the simulation without GM (Case 4a; horizontal diffusion + Ekman; solid blue line), with differences reaching 5–7 °C (see Fig. c). In contrast, adding convective adjustment on top of GM (Case 7a; solid red line) produces only negligible additional changes (Fig. c). This indicates that, under the conditions tested here, the inclusion of GM substantially modifies vertical stratification such that the contribution of convective adjustment to deep layer temperatures is minor.

We next consider Case 4b (horizontal diffusion + Ekman + convective adjustment; dash-dotted blue line). Compared to Case 4a, convective adjustment reduces the deep layer temperatures (Fig. b), but this cooling is weaker than that produced by GM only (Fig. c). Still, this highlights that the effects of GM and convective adjustment are not simply additive. To further isolate the role of GM when convective adjustment is already active, we compare Case 4b with Case 7a. This reveals that the inclusion of GM, which drives eddy-induced heat redistribution, produces additional mid-latitude cooling of 2–3 °C (Fig. d). This shows that GM has a distinct influence on the deep ocean even in the presence of convective adjustment.

To test sensitivity to the strength of the GM scheme, we performed Case 7b, in which the GM transfer coefficient is increased to 2000 m² s⁻¹ (compared to 1000 m² s⁻¹ in Case 7a). Due to the increased coefficient, mid-latitude cooling reaches 3–4 °C (Fig. d), consistent with stronger eddy-induced redistribution of heat. This could be particularly relevant for slowly rotating planets, for which mixing coefficients could be substantially larger e.g.,. This will be investigated in future work.

Overall, these results show that convective adjustment alone does not reproduce the equilibrium climate obtained when GM is included. Even when convection is active, adding GM leads to systematic cooling of the deep ocean at mid-latitudes, whereas adding convective adjustment to a configuration that already includes GM produces only minor additional changes.

This is further emphasised in another set of simulations to assess whether an enhanced horizontal diffusion and convective adjustment can substitute for GM. For this, we compared a -style configuration (horizontal diffusion coefficient of 25 000 m² s⁻¹, Ekman transport and convective adjustment) with the recommended configuration of the present model including GM (Case 7b). Our findings show that even with substantially enhanced horizontal diffusion and active convection, the absence of GM leads to systematically warmer mid-latitude deep-layer temperatures. This comparison is discussed in detail in Appendix .

To further contextualise the present model, we compare the OHT structure obtained using the configuration of . Figure  presents total and decomposed meridional OHT profiles for three configurations, allowing us to isolate the effects of the Sverdrup balance, GM transport and updated parameter choices.

Panel (a) reproduces the oceanic configuration as closely as possible, with horizontal diffusion (coefficient of 25 000 m² s⁻¹), Ekman transport (without the Sverdrup balance) and convective adjustment. In this configuration, the OHT exhibits a pronounced hemispheric asymmetry, discussed in detail in Sect. . Interestingly, the OHT profile in (see Fig. ) is quite symmetric, despite the absence of Sverdrup balance in that study. This could point towards a decreased sensitivity of the atmospheric model to surface temperature variations in that version of the GCM. Correspondingly, it implies that the current atmospheric model is either sufficiently or overly sensitive to surface temperature variations. While investigating the related sensitivity is beyond the scope of this work, its physical reasoning is explained in Sect. .

Panel (b) shows the same Codron-style configuration but with the Sverdrup balance enabled. This largely restores hemispheric symmetry in the OHT, showing its negative-feedback-induced stabilising role (see Sect. ). However, despite this improvement, the amplitude of tropical Ekman transport remains suppressed relative to AOGCM benchmarks e.g.,. This indicates that the Sverdrup balance alone is insufficient to recover a realistic OHT structure when GM transport is absent.

Panel (c) presents the recommended oceanic configuration of the present model, which uses a reduced horizontal diffusion coefficient (8000 m² s⁻¹), GM transport (2000 m² s⁻¹), Ekman transport with the Sverdrup balance, and convective adjustment. Here, the tropical OHT peak – primarily driven by Ekman transport – reaches amplitudes closer to those in fully coupled AOGCM simulations (e.g., Fig. 7 of ; hot-state solutions in Fig. 2 of ) than those obtained either in Fig. 4 or in the setups with the current model. This increase arises since GM transport modifies the temperature contrast between the surface and deep layers, thereby affecting the strength of Ekman transport (Eq. ). Likewise, the Ekman-driven mid-latitude trough is more pronounced due to GM transport in panel (c), resembling the amplitude of the Eulerian transport component in AOGCM simulations (Marshall et al., 2007; Fig. 7). Compared to the ≈30° peak in Fig. 4 of , the latitude of the diffusive transport peak is also shifted poleward to approximately 45°, in closer agreement with . Its peak amplitude (around 0.6 PW) is provided solely by horizontal diffusion in panels (a) and (b). In contrast, the present model decomposes diffusive transport into its horizontal diffusion and GM transport components, yielding a slightly larger combined diffusive peak of ≈0.75 PW in panel (c).

The tropical OHT peak is also sensitive to the system's climatic attractor. In panel (c), which corresponds to a “hot state” see with no sea ice, the tropical OHT peak is approximately 1.75 PW, consistent with the hot state solution of Fig. 2 obtained with the MITgcm. In contrast, in simulations tuned to reproduce an Earth-like sea-ice distribution (e.g., Fig.  in this work), we obtained peak transports of 2–3 PW, similar to the warm state attractor in .

These comparisons demonstrate that the combined inclusion of the novel features of the model: Sverdrup balance, GM transport, and updated parameter choices, leads to an improvement over . The resulting OHT structure more closely resembles that of fully dynamic AOGCMs, despite the relative simplicity of our model.

Previous aquaplanet simulations with obliquity using the Generic-PCM (e.g., ) revealed a persistent pronounced hemispheric asymmetry in surface temperature and cloud cover, even after year-long averaging. In , simulations were initialised starting from the Vernal Equinox and without OHT. The resulting asymmetry was attributed to an initial hemispheric bias that triggered a net north–south atmospheric heat transport, allowing one Hadley cell to dominate over the other. This induced a feedback that persisted throughout the simulation.

We propose an additional explanation involving the role of OHT. Our oblique aquaplanet simulations show that when OHT is disabled (blue line in Fig. ), the hemispheric bias imprinted at model initialisation is not corrected by the system. In the absence of OHT, there is no mechanism – apart from atmospheric transport – to redistribute excess energy across hemispheres, triggering a cascade of feedbacks. The initially warmer Northern Hemisphere delays sea ice growth, while the Southern Hemisphere cools rapidly due to atmospheric transport acting on short timescales. This cooling is amplified by the ice–albedo feedback, leading to early and extensive sea ice formation, which locks in a strong hemispheric asymmetry. Atmospheric heat redistribution alone is insufficient at evening out hemispherical differences – and can even amplify them – through feedbacks involving the cross-equatorial Hadley cell, which develops with ascending motion in the warmer hemisphere. The associated cloud cover and moisture content further modulate the radiation budget, reinforcing the asymmetry.

When OHT is on (red line in Fig. ), the ocean helps smooth out temperature gradients, mitigating the imprint of the initial bias. At model start, heat stored in the ocean is transported from lower to higher latitudes in both hemispheres through the mechanisms discussed in Sect. . This redistribution limits excessive sea ice formation in the South and prevents the North from staying anomalously warm. The ocean circulation forced by the cross-equatorial Hadley cell will also transport heat across the equator towards the cold hemisphere, with opposite feedbacks from the ones associated with atmospheric transport. In general, strong persistent hemispheric asymmetries in aquaplanet setups are unphysical and typically reflect initial condition biases and/or numerical artefacts. The key takeaway is that both oceanic and atmospheric heat transport are essential for suppressing such asymmetries and maintaining a physically realistic climate state.

In our model, this stabilising effect of OHT is largely driven by the wind-curl-sensitive Sverdrup balance scheme implemented near the equator (Sect. ). To assess the impact of this scheme, we conducted two simulations with Sverdrup transport disabled, one initialised from the converged Case 7 (non-oblique) state, and the other from an isothermal 290 K start to check for dependence on the start state.

Without the Sverdrup balance, near-equatorial OHT responds purely through frictional transport (see Eq. ). In this case, the cross-equatorial OHT occurs in the same direction as the surface wind. This generates an SST asymmetry (Fig. a) that shifts the ascending branch of the Hadley cell, thereby amplifying the cross-equatorial wind anomaly (Fig. a). This leads to a strong hemisphere-wide dichotomy of temperature anomalies (Fig. b). Moreover, our two Sverdrup-off experiments converge to opposite asymmetric states, illustrating this bistability (Fig. a).

With the Sverdrup balance enabled (Case 7), the near-equatorial transport responds to the wind stress curl rather than its frictional meridional component. For hemispherically symmetric forcing, the wind stress curl remains symmetric about the equator. Importantly, the resulting cross-equatorial OHT occurs opposite to the surface wind direction, introducing a negative feedback that suppresses growing inter-hemispheric asymmetries. As a result, the coupled system converges to a more symmetric equilibrium, both for SST (Fig. a) and net surface winds (Fig. b). Consequently, the inclusion of the Sverdrup balance in our model plays a crucial role in preventing the locking-in of hemispheric biases and maintaining a physically consistent climate.

With further experiments, we found that enabling OHT also enhances numerical stability in the Generic-PCM. In the OHT-off configuration, the model frequently encountered dynamical instabilities during long integrations. These usually manifest as spurious temperature dips near the surface or at the top of the atmosphere, often triggering radiative transfer scheme failures (temperatures dropping below the lower bounds of the lookup tables), causing the model to crash. These issues are largely absent when OHT is enabled. A likely explanation is that OHT efficiently smooths meridional temperature gradients – particularly at smaller scales due to the diffusive components – thereby reducing the likelihood of instabilities and excessive vertical motions. These OHT-smoothened temperature fields may also improve numerical stability by reducing sharp vertical gradients that can otherwise trigger Courant–Friedrichs–Lewy (CFL) violations in the model's dynamical core.

One approach in low-complexity ocean modelling is the q-flux method, where OHT is prescribed as a fixed meridional heat flux (q-flux) based on observational data or AOGCM simulations. While this avoids the computational cost of explicitly solving ocean dynamics, it requires an initial, expensive reference simulation to determine the appropriate OHT. In the context of exoplanets, where observational constraints are limited and reference dynamical ocean simulations are typically unavailable with some exceptions: for e.g.,, a physically motivated q-flux cannot be prescribed. More critically, if an OHT is imposed rather than emergent, it cannot dynamically respond to changes in the atmosphere, ocean, or external forcing. As a result, key climate feedbacks – such as adjustments in meridional heat transport due to sea ice growth, atmospheric circulation shifts, or changes in composition (e.g., chemistry, greenhouse gases) – are not captured self-consistently.

In contrast, our dynamical slab ocean model allows for an emergent OHT based on ocean-atmosphere interactions. This is particularly important in cases where strong feedbacks develop – such as the ice-albedo effect in our simulations or the water vapour feedback in – and more broadly because atmospheric and oceanic heat transports are strongly coupled via surface winds. We note, however, that the present model is primarily designed for Earth-like rotation rates and parameter regimes. For instance, our Sverdrup balance may overestimate transport for slowly rotating planets (Sect. ), and diffusion coefficients are calibrated using Earth-rotation aquaplanet benchmarks (see Sects.  and ). Given this, our approach currently shares with q-flux methods a reliance on well-characterised regimes. Nevertheless, a key distinction is that even within these regimes, OHT in our model is not prescribed but responds interactively to changes in atmospheric forcing and climate state. Extending the framework to other planetary contexts is a natural direction for future work.

Figure a explores model sensitivity to the GM and horizontal diffusion coefficients. In the OHT-off simulation (blue), surface temperatures at high northern latitudes (60–90° N) are up to 15 °C colder than the NCEP/NCAR reanalysis (black dashed line). Introducing OHT by increasing the GM coefficient (green and red lines, for 1000 and 2000 m² s⁻¹, respectively) and/or varying the horizontal diffusion (8000 to 25 000 m² s⁻¹) warms the Arctic modestly, raising temperatures by about 3–4 °C. However, this is insufficient to eliminate the persistent cold bias. Seasonal analysis reveals that while summer Arctic temperatures fall within expected bounds (-10 to 0 °C), average winter temperatures plunge to ≈ −50 °C – approximately 20 °C colder than observed. This suggests that a more dominant process is responsible, likely involving a combination of atmospheric transport inefficiencies, radiative imbalances, and/or model tuning choices such as surface albedo.

Figure b illustrates the model's sensitivity to sea ice and snow albedo, which is reduced from 0.65 to 0.60 and 0.55 for the OHT-on simulation. Lowering the albedo does mitigate the Arctic cold bias, with a stronger warming effect than seen in the GM experiments. However, this comes at the cost of Southern Hemisphere sea ice almost completely disappearing, while concurrently reducing annual global sea ice extents to unrealistic levels (10 million km²), introducing a new discrepancy. The systematic Southern Hemisphere warm bias also hints at a hemispheric imbalance, where the (orbital-parameter-induced) excess heat in the south is not being effectively redistributed to the Northern Hemisphere, despite the heat transport provided by the ocean. Additionally, the atmospheric model of the GCM may under-represent poleward moist static energy transport, particularly via latent heat. Together, these factors suggest a complex interplay between radiative, oceanic, and atmospheric processes that warrants deeper investigation. Nevertheless, we reiterate that these discrepancies remain within acceptable bounds, given that our model is not built with the intention of precisely reproducing modern Earth's climate but rather that of exoplanets, where observational constraints are far more uncertain.

We summarise below the key developments of the present model relative to earlier implementations that enhance its suitability for exoplanet and paleoclimate applications.

Two of our improvements are particularly relevant for simulating M-star planets, which are of high interest given their abundance in the galaxy and their observational accessibility. First, the inclusion of a spectrally dependent sea ice and snow albedo (Sect. ) enables more realistic radiative feedbacks on planets receiving non-solar spectra. For M-dwarfs, broadband albedos of snow and ice can be 25 %–55% lower than for G-type stars , making this refinement essential for accurate climate modelling. This is also critically important to better interpret observational data from the JWST and in the future, the ELT.

Second, the introduction of GM transport represents more than a numerical improvement. Through its transfer coefficient, the GM scheme acts as a tuning parameter that controls the effective scale of mesoscale eddies (see Sect. ), enabling exploration of different rotational regimes, notably slow rotators such as TRAPPIST-1e (rotation period 6.1 d) and Proxima b (11.2 d), for which mixing is expected to differ substantially from Earth e.g.,. In earlier models lacking GM , no consistent framework existed to account for such regime-dependent mixing.

Another strength of the present framework is its flexible architecture. Individual oceanic processes can be selectively (de)activated through configuration flags, allowing users to isolate processes, depending on their need. Moreover, the modular design enables technically straightforward integration of additional processes, such as a simplified sea-ice drift scheme (see Sect. ), also important for exoplanet applications .

Finally, the present ocean model is fully parallelised, in contrast to the implementations of and . This enables efficient long integrations and large ensemble simulations (see Appendix  for more details), which are essential for exoplanet and paleoclimate studies where observational constraints are sparse.

One of the driving applications for developing the dynamical slab ocean model is to simulate the climates of terrestrial exoplanets. Given that the model is flexible, relatively easy modifications can be made. For example, some exoplanets are expected to experience non-negligible geothermal heat fluxes – this additional energy source can be incorporated in the model as a constant or spatiotemporally varying ground-heating term (see Eq. ). Internal tidal heating may also represent an additional and potentially dominant heat source for some close-in terrestrial exoplanets e.g.,. Furthermore, tidal forcing of the ocean can influence the large-scale circulation and climate, particularly for aquaplanets orbiting low-mass stars, where strong tides can significantly influence ocean mixing and heat transport e.g.,. These additional physical processes represent promising future directions.

Moreover, many of the currently known terrestrial exoplanets are expected to be in synchronous rotation, particularly those orbiting close to M-dwarf stars. In such cases, the planet's rotation period equals its orbital period – typically ranging from 5 to 15 Earth days for temperate planets. This can place these planets in dynamical regimes distinct from Earth-like rotators for e.g., see. Most climate model parameters, including those governing subgrid-scale mixing, are traditionally calibrated for Earth-like rotation rates. For instance, the diffusion coefficients discussed in Sect.  were selected to match the meridional ocean heat transport of an Earth-like aquaplanet . However, the reduced Coriolis parameter f on slowly rotating planets will lead to a larger Rossby deformation radius (since LD∼g′H/f, where g′ and H are the reduced gravity and scale height respectively), which sets the typical scale of baroclinic eddies, from 10 to 100 km on Earth's ocean. This suggests that eddy-driven mixing processes may differ substantially on such climates (see and ). Though our model does not include the density-driven component of the MOC, the GM scheme modifies its temperature-driven part, and leads to high-latitude surface warming and low-latitude depth cooling. This is qualitatively similar to the behaviour reported by , who showed that slower planetary rotation rates enhanced MOC strength and led to comparable thermal redistribution. This is a strong motivation for us to tune the GM coefficient in future exoplanet implementations of the model.

A related limitation arises from our assumption of a globally uniform, fixed mixed-layer depth (MLD). In reality, the MLD varies with location and season due to wind forcing, ocean currents, and surface heating e.g.,. also showed that mixed layer seasonality increases with planetary obliquity. The fixed MLD approach simplifies calculations of ocean heat transport, but can introduce biases. First, regions that should have a deeper MLD, such as high-latitude oceans during winter, may exhibit larger or faster surface temperature variations due to an artificially small heat capacity. Conversely, regions that should have a shallower MLD, such as tropical oceans, may experience overly dampened temperature variations. This limitation could impact the accuracy of (short-timescale) climate feedbacks, particularly the ice-albedo feedback, since a dynamically varying MLD would allow for more realistic oceanic heat uptake and redistribution. Choosing a representative MLD is particularly challenging for exoplanets, where no observational constraints exist – akin to the challenge of prescribing a q-flux. A promising direction for future work is to adopt an “Ekman mixed-layer ocean” scheme, which uses multiple slab layers and includes explicit vertical mixing. This approach, demonstrated by – who combined it with the Ekman transport parameterisation of – may offer a more physically robust framework than using a fixed-depth single-layer ocean.

Our model includes salinity solely as a modifier of the freezing point of seawater. This is an important modification as a depressed freezing point inhibits sea ice formation – leading to reduced global sea ice cover with increasing salinity. demonstrated that higher salinities tend to warm planetary climates – offering a potential solution to the Faint Young Sun Paradox for Archean Earth – and attributed this warming primarily to changes in density-driven ocean dynamics rather than to freezing point depression. Building on this, found that while salinity has a strong, nonlinear impact on G-star planets, leading to abrupt sea ice loss and surface warming, its effect on M-dwarf planets is more gradual, with minimal associated warming. Our model does not represent the dynamical effects of salinity – such as its role in setting ocean density or driving thermohaline circulation. As highlighted by , this can lead to an underestimation of convective exchanges in high-latitude regions and inadequate cooling of the deep ocean. The absence of salinity also prevents the model from capturing salt rejection during sea ice formation, a process that increases local water density and promotes vertical mixing. While density-driven circulation due to salinity is important in shaping OHT e.g.,, its inclusion would substantially increase model complexity and computational cost. In this context, our GM transport scheme provides an intermediate complexity alternative balancing realism and efficiency. For exoplanet studies, where the priority lies in capturing first-order climate dynamics rather than detailed regional processes, this simplification remains justified. Nonetheless, future extensions of our model could explore ways to incorporate simplified density-driven processes without the overhead of full ocean dynamics.

The model does not simulate sea ice drift, a simplification made to reduce computation time. On modern Earth, this omission can lead to unrealistically high sea ice accumulation in the Arctic (as seen from our model data in Fig. ), as ice is unable to be transported out of the region and instead thickens and expands in place . While this may be a secondary effect under present-day Earth conditions, sea ice dynamics has been shown to play a critical role in shaping planetary climate regimes. They are essential for simulating transitions between stable climate states on Earth and exoplanets. For example, dynamic sea ice facilitates equatorward transport, promoting Snowball Earth initiation and enhancing the instability of tropical sea ice margins . These processes may be even more important on synchronously rotating planets, where strong day–night contrasts can interact with sea ice transport to produce qualitatively different climate outcomes . Future work is needed to assess whether incorporating sea ice drift is feasible within the framework of our model without incurring prohibitive computational costs.