A nonlinear, steady-state, one-dimensional conductive heat-transfer equation is solved as follows: 1∂∂zk(T,z)∂T∂z+A(z)=0, where T is temperature [K], k(T,z) is a temperature-dependent thermal conductivity [Wm-1K-1], and A(z) is radiogenic heat production [Wm-3 in the governing equations (model input values are reported in µWm-3)], both expressed as functions of depth. Depth z is positive downward, and radiogenic heat production A is defined as a positive volumetric heat source. Accordingly, upward exhumation corresponds to a negative vertical material velocity in the transient models of Sect. 4.

Temperature-dependent thermal conductivity is modelled following Jaupart et al. (2016), with an additional temperature-dependent radiative component at temperatures >1000 K (Jaupart and Mareschal, 2005). The governing equations are: 2klat(T,z)=2.26-618.251T+ko(z)355.576T-0.30247, and 3krad(T)=0.37×10-9T3,T≥1000K;krad(T)=0,T<1000K, where ko is the thermal conductivity of the rock under surface conditions, which differs between lithological layers (Table 1), klat is the lattice (non-radiative) contribution and krad the radiative contribution, which is negligible below 1000 K. The total thermal conductivity is therefore: 4k(T,z)=klat(T,z)+krad(T).

Equation (1) is solved using Dirichlet boundary conditions with fixed temperatures at the surface and the base of the lithosphere. The model allows an arbitrary number of layers, each with distinct ko and A. To handle abrupt changes in properties, a finite-difference “half-station” scheme is employed (Hindle and Besson, 2023). A tridiagonal matrix algorithm combined with fixed-point iteration solves the nonlinear system: 5(K[k]T)i=ki+1/2(Ti+1-Ti)-ki-1/2(Ti-Ti-1)Δz2. The geological model contains abrupt lithological contrasts in conductivity and heat production, so the conservative operator ∂/∂z (k∂T/∂z) is discretized directly at half stations. For a uniform grid, this gives Eq. (5), where K[k] denotes the discrete conservative conduction operator and ki+1/2 and ki-1/2 are conductivities evaluated at the cell faces.

No numerical derivative of k is required. This matters because first expanding the operator as k∂2T/∂z2+(∂k/∂z)(∂T/∂z) implicitly assumes that k is differentiable, whereas at an abrupt lithological boundary ∂k/∂z is not an ordinary function. The half-station formulation instead works with conductive fluxes across cell faces and preserves the conservative structure of the equation. The same operator is used in the transient calculations of Sect. 4.

Several scenarios were tested to explore the sensitivity of crustal temperatures to key parameters such as lithospheric thickness and upper-crustal radiogenic heat production.

A nonlinear, transient, one-dimensional advective–conductive heat equation with internal radiogenic heat production is solved as: 6ρc∂T∂t+vz∂T∂z=∂∂zk(T,z)∂T∂z+A(z), where z [m] is positive downward, so that upward exhumation corresponds to vz<0 [ms-1]. T is temperature [K], k(T,z) the temperature-dependent thermal conductivity [Wm-1K-1], A(z) the radiogenic heat production [Wm-3] with model input values are reported in µWm-3, ρ the density [kgm-3] and c the specific heat capacity [Jkg-1K-1].

Advection and conduction are treated by operator splitting on a fixed grid. The prescribed exhumation displacement is accumulated in time; whenever it reaches one grid interval Δz, the temperature field and the associated material properties, such as reference thermal conductivity, radiogenic heat production, density and specific heat capacity, are translated upward by one node. Material leaving the upper boundary is removed, and material carrying the properties of the lowermost layer (Table 2) is introduced at the base. We denote by Tn the temperature field after this remapping step. Advection is therefore represented by fixed-grid material remapping rather than by direct finite-difference approximation of vz∂T/∂z. Equation (6) states the physical conservation law, whereas the remapping procedure specifies how its material-transport component is represented numerically.

The temperature-dependent conductivity is lagged as follows: 7akn=k(Tn,z), then the nodal update: 7ρicp,iTin+1-T̃inΔt=(K[kn]Tn+1)i+Ai,8(C-ΔtLn)Tn+1=CT̃n+ΔtA with 8aC=diag(ρicp,i),Ln=K[kn]. Here T̃n is the temperature field immediately after the remapping step, C=diag(ρicp,i) is the volumetric heat-capacity matrix and Ln is the tridiagonal conservative conduction operator assembled from the half-station conductivities of Eq. (5), evaluated at the preceding temperature state. Volumetric heat capacity is therefore retained explicitly and varies between layers (Table 2). The temperature field is advanced with backward Euler while the conductivity is lagged, so the scheme is linearly implicit with lagged coefficients rather than a fully nonlinear implicit solve. Dirichlet conditions are imposed at the top and base of the column by modifying the corresponding rows of C and Ln. The system is solved at each timestep with a tridiagonal matrix algorithm.

To model the transient effect of exhumation on geotherms in a dynamically evolving orogenic belt, the ideal initial condition would represent the pre-collision state. One approach is to consider the present-day cratonic India as an analogue for the pre-collision Indian margin and to simulate subduction of the Indian passive margin and its sedimentary cover beneath Kohistan from 50 Ma onward, thereby increasing crustal thickness. However, uncertainties in the dynamic and kinematic conditions between 50 and 10 Ma before the onset of rapid exhumation make this approach challenging (Zeitler, 1985). It is also likely that, over ∼35 Ma, crustal geotherms re-equilibrated to a near-steady-state condition (Fairley, 2016).

Zircon fission-track data (Zeitler, 1985) show very young cooling ages (<5 Ma) in the NPM and Karakoram, versus >10 Ma in Kohistan, indicating rapid exhumation (∼7 mmyr-1) (Fig. 5a). Alternative interpretations suggest slower exhumation between 50–10 Ma followed by constant rates of 3–4 mmyr-1 over the last 10 Ma (Whittington, 1996). A recent petrochronologic and thermal-modelling study suggested an ultrafast exhumation pulse in the NPM (∼9–13 mmyr-1 beginning at ∼1 Ma), preceded by sustained rapid exhumation of 2–5 mmyr-1 (Guevara et al., 2022). However, this ultrafast exhumation is probably localized to the core of the NPM because river-sediment flux estimates suggest 3–5 mmyr-1 as the regional exhumation rate (Clift et al., 2022).

For this work, we do not consider the ultrafast exhumation scenario and test only the effect of varying regional exhumation rates and RHP on temperature and surface heat flow. Because material is continuously removed at the surface and introduced at the base (Sect. 4.1), the column represents a fixed window through which solid material passes rather than a deforming block: its total thickness remains 25 km, while the fraction occupied by the upper, more radiogenic layer decreases and that occupied by the lower layer increases as exhumation proceeds. Over 10 Ma, this corresponds to 10, 20 and 30 km of throughflow at 1, 2, and 3 mmyr-1, respectively, so that in the two faster scenarios the original upper layer is entirely exhumed before the end of the simulation and the elevated temperatures are sustained by advection rather than by the initial heat-production architecture.

The transient model is intentionally restricted to the uppermost 25 km of the crust and is not a transient simulation of the entire lithosphere. Its purpose is to isolate the sensitivity of the geothermally relevant upper crust to sustained exhumation, at a resolution and computational cost that permit systematic variation of exhumation rate and RHP. The base of the domain is fixed at 800 °C, corresponding to the metamorphic conditions inferred beneath the NPM (Crowley et al., 2009; Guevara et al., 2022), and is consistent with the 578–686 °C obtained at 20 km in the steady-state models of Sect. 3. Holding this boundary fixed assumes that the thermal structure below 25 km is unaffected by exhumation. In a fully coupled calculation, the deeper geotherm would also be advected upward, supplying material warmer than 800 °C to the base of the window; the transient warming reported here is therefore conservative in this respect.

Figure 5b and Table 3 summarize the effects of low and high RHP scenarios with exhumation rates of 1–3 mmyr-1. Geotherms after 10 Ma show significantly higher temperatures than the initial geotherms. For the high RHP models at 3 mmyr-1, the geotherms invert at a depth of 20 km depth. Increasing exhumation rates in low RHP models by 1 mmyr-1 produces geotherms comparable to high RHP models. At 10 km depth, low RHP geotherms increase by a factor of 1.22, while high RHP models increase by 1.14 per 1 mmyr-1 increase in exhumation.

Surface heat flow also rises with exhumation. For low RHP models, surface heat flow increases from 75 mWm-2 (initial) to 100, 160, and 220 mWm-2 for exhumation rates of 1, 2, and 3 mmyr-1, respectively. High RHP models show an increase from 110 mWm-2 to 135, 180, and 250 mWm-2 for the same exhumation rates (Fig. 5c). The highest values in the transient end-member scenarios combining sustained 3 mmyr-1 exhumation over 10 Ma with high RHP represent exploratory upper bounds rather than expected regional conditions, and could apply spatially only to the localized, rapidly exhuming core of the NPM syntaxis, consistent with the spatial confinement seen in the 2D results (Fig. 7).

Because the basal temperature is held fixed while internal heat production continues, material advected upward can exceed the imposed basal value, so that temperatures within the column locally exceed 800 °C and the profile inverts below the resulting maximum. The inverted geotherms obtained in the high-RHP, 3 mmyr-1 case are therefore a joint consequence of the prescribed basal condition and the imposed heat production, and mark the limit of validity of the fixed-boundary, fixed-RHP formulation rather than a predicted physical state.

The Péclet number provides a dimensionless measure of the relative importance of advective and conductive heat transport and can be expressed as Pe=vL/κ, where v is the exhumation velocity, L a characteristic length scale, and κ the thermal diffusivity. For the 25 km model domain and κ≈10-6 m2s-1, exhumation rates of 1, 2 and 3 mmyr-1 yield Pe values of approximately 0.8, 1.6 and 2.4, respectively. Thus, advection is comparable to conduction at 1 mmyr-1 and increasingly important at higher exhumation rates, consistent with the progressive amplification of the geotherms.

The tectonic framework of the Himalaya–Kohistan–Karakoram system exerts a profound influence on its present-day thermal state. Subduction and imbrication of the Indian Plate have produced substantial crustal thickening, while subsequent exhumation has advected heat toward the surface. These processes act in tandem: crustal thickening increases the vertically integrated radiogenic heat production of the crust (Jaupart et al., 2016), whereas exhumation rapidly transports hot rocks upward, generating strong near-surface anomalies (Jamieson et al., 2002). Additionally, thickened crust fosters partial melting at depth, with melts emplaced as plutons in the upper crust (Jamieson and Beaumont, 2011). Collectively, these processes contribute to the thick crust, elevated crustal temperatures, shallow Curie depths (∼580 °C isotherm), and anomalously high background heat flow commonly observed in active orogenic belts (Gao et al., 2017).

Developing reliable lithospheric thermal models therefore requires robust constraints on crustal structure and physical properties. In stable cratonic settings, this is feasible because steady-state models can be anchored to deep borehole heat-flow data, which minimize transient near-surface effects (Fuchs et al., 2022). In contrast, active orogens such as the Himalaya lack reliable surface heat-flow datasets and suffer from poorly constrained geophysical profiles at depth. The present study addresses this challenge by adopting a suite of models with variable parameters, thereby bracketing the plausible thermal states of the lithosphere rather than committing to a single deterministic solution.

Within thermal models of the continental crust, RHP is widely regarded as the dominant parameter controlling crustal geotherms (Hasterok and Chapman, 2011). However, its vertical distribution remains ambiguous. It is generally accepted that lower-crustal melting and melt extraction redistribute heat-producing elements upwards in the crust, leading to depletion in the lower crust and enrichment in the upper layers (McKenzie and Priestley, 2016). At mid-crustal levels, this redistribution is neither complete nor uniform. U and Th are commonly co-enriched in evolved granitic melts and are broadly positively correlated across the lithologies in the study area. Both contribute significantly to RHP, and both may be retained variably in resilient accessory phases (e.g., monazite, zircon, allanite) during near- or sub-solidus melting (Alessio et al., 2018; Yakymchuk and Brown, 2019; Weller et al., 2020). Migmatitic gneisses exhumed from mid-crustal levels in the NPM nonetheless show high Th/U ratios relative to the associated leucogranites (Anees et al., 2024), illustrating that partitioning behaviour varies among lithologies within a single terrane and confirming the absence of any systematic negative correlation of RHP with increasing metamorphic grade (Hasterok et al., 2018).

The present modelling approach provides a critical evaluation of these hypotheses. Assigning surface-derived RHP values uniformly through the crust produces unrealistically hot geotherms (Figs. 4 and S2), with Moho temperatures far exceeding petrological and geophysical constraints. This strongly suggests that RHP diminishes with depth in active orogens, likely through fractionation, melt extraction, and redistribution processes. At the same time, localized upper-crustal enrichment by pluton emplacement may sustain high surface values. Thus, the results argue against simplistic uniform-depth assumptions and instead favour models that incorporate depth-dependent variability in RHP. The multi-scenario approach constrains the direction of the required depletion but not its precise depth distribution, which remains the principal unresolved uncertainty of the model.

The similarly high differentiation indices of the NPM and the Bundelkhand craton contrast with their markedly different average crustal heat production values, which are considerably higher in the modelled NPM value (1.5 µWm-3) than in Bundelkhand (0.41–0.68 µWm-3). This difference reflects composition rather than thickness alone: crustal thickening by imbrication and under-thrusting in the NPM stacks fertile Indian upper- and middle-crustal material, raising the mean volumetric heat production of the resulting column. For a given mean volumetric RHP, increasing crustal thickness increases the vertically integrated radiogenic heat production and therefore the crustal contribution to surface heat flow, so a thickened and compositionally fertile column produces elevated temperatures at mid- to lower-crustal depths. These conditions may promote crustal melting and the redistribution of heat-producing elements toward the upper crust, potentially increasing the differentiation index without substantially altering bulk mean crustal heat production if the elements remain within the crustal column. This redistribution may also moderate lower-crustal temperatures (Perry et al., 2006). This is also corroborated by our modelling results, where higher RHP (≥2 µWm-3) layers in the mid-crust generated exceptionally hot geotherms (Figs. 4 and S2).

Recent thermomechanical models further emphasize the role of radiogenic enrichment in the upper crust. Fan et al. (2024) demonstrated that sustained ultra-high-temperature metamorphism in the Greater Himalayan Crystalline Complex was primarily driven by elevated radiogenic heat production (∼4 µWm-3) in a partially molten upper crust, with mechanical heating playing only a transient role during the early stages of collision. This highlights that crustal differentiation, whereby heat-producing elements are preferentially concentrated in the upper crust through partial melting and magmatic segregation, can maintain high geotherms over tens of millions of years. Such findings are consistent with our results, where high DI in the NPM and the Karakoram reflect strong upper-crustal enrichment in radiogenic elements, enhancing the thermal gradients.

1D modelling results indicate that surface heat flow is strongly controlled by the thickness and intensity of heat-producing layers in the upper crust. While transient 1D models emphasize the influence of exhumation, 2D models suggest that its lateral impact is limited compared to radiogenic heat production (RHP). In the NPM, a high-temperature thermal regime results from the combined effects of exhumation and RHP, producing elevated near-surface geothermal gradients. Consequently, the thermal structure beneath rapidly exhuming regions, particularly the Nanga Parbat Massif, cannot be adequately described by purely conductive models. Péclet numbers of approximately 1.6–2.4 in the 1D rapid-exhumation scenarios indicate that advective transport is comparable to or greater than conductive heat transfer and therefore contributes substantially to the upper-crustal thermal structure. Nevertheless, radiogenic heat production remains an important internal heat source controlling the regional thermal field.

The 2D models indicate that topographic relief modulates the near-surface conductive temperature field, while the concentration of hot springs in deeply incised valleys is consistent with additional structural and hydrogeological controls that are not represented in the model. In addition, unlike purely conductive heat transport, topography-driven deep groundwater flow interacting with heat-producing rocks promotes advective heat transfer to the surface. This advective process can transfer heat to the surface more efficiently and rapidly than conduction alone, particularly in evolving orogens (Chamberlain et al., 1995; Wanner et al., 2020). While extreme near-surface gradients have independently been inferred at Nanga Parbat from evidence of shallow boiling hydrothermal systems and Pleistocene anatexis (Chamberlain et al., 1995; Crowley et al., 2009), these likely represent localized anomalies rather than background conditions.

The Himalayan Geothermal Belt hosts several high-temperature hydrothermal systems in which heat is convected by deep circulation of meteoric water. At Yangbajing in Tibet, a shallow reservoir reaches 150–165 °C at only 180–280 m depth, while a deeper reservoir at 950–2000 m attains 250–329 °C (Fan, 2002; Xue et al., 2023). In the Puga Valley of Ladakh, reservoir temperatures of ∼200–250 °C are inferred at depths of 1–2 km (Shanker et al., 2000; Harinarayana et al., 2006). Our models for northern Pakistan indicate modelled surface heat flow locally exceeding 120–180 mWm-2 and modelled temperatures above 200 °C at approximately 3 km depth (below surface) in major valleys of the NPM and parts of the Karakoram. We emphasise that, unlike the drilled and geophysically constrained values quoted above for Yangbajing and Puga, these are scenario-based model outputs derived without calibration against measured subsurface temperatures. Subject to that distinction, they still indicate a thermal regime comparable to medium- to high-temperature geothermal systems and underline its relevance within the belt. Under the more conservative exhumation and RHP scenarios, modelled isotherms exceeding 100 °C occur at depths of 1200–2500 m (below surface) in the valleys.

A possible indication of fluid interaction with uranium-bearing lithologies comes from radon measurements in regional hot spring waters. Some recent publications have reported spatially variable radon concentrations, with the highest value of 304 BqL-1 at the Tattapani spring at Raikot Bridge on the western flank of the NPM (Ullah et al., 2022; Muhammad and Haq, 2023). Because 222Rn derives from the 238U decay series and has a half-life of only 3.8 d, such values indicate possible contact with uranium-bearing rock along the shallow portion of the flow path. However, they provide no direct constraint on the contributions of Th or K (which also contribute to RHP) and no information on a deeper heat source. Both studies were undertaken for health risk assessment rather than geothermal characterisation. A dedicated noble-gas survey, in particular of 4He/3He, would be considerably more diagnostic, since radiogenic 4He accumulates directly from U and Th decay and would test the crustal origin of the heat inferred from the low 3He/4He ratios reported for the belt (Hochstein and Zhongke, 1995).

The results indicate favourable thermal conditions, which constitute one of the major components of geothermal resource potential. Moving from thermal favourability to a resource estimate, and ultimately to viable exploration targets, requires additional characterisation of permeability, fracture-network connectivity, in situ stress and hydrogeological conditions, none of which are addressed by the present models. Nonetheless, the deeply incised valleys of the NPM and Karakoram represent areas of elevated thermal favourability that warrant further site-specific investigation.

The thermal models presented here provide first-order estimates of crustal geotherms and surface heat flow, while accounting for uncertainties arising from limited data and heterogeneous parameters. Boundary conditions represent a further source of uncertainty that is only partially assessed. The sensitivity test presented in Sect. 3.2.1 varies the depth of the basal boundary but not the temperature prescribed at it or the type of condition imposed, and it is a local rather than a global test in the sense of Degen et al. (2025): parameters are varied individually about a reference configuration, so correlations between the basal boundary and other model parameters are not resolved. A systematic assessment, varying LAB depth, LAB temperature and boundary type jointly, would establish more firmly which features of the predicted geotherms are independent of these choices. In these models, intra-layer RHP values are assumed to be constant. However, as noted in Sect. 7.1, assigning surface RHP values to mid-crustal layers leads to unrealistically high temperatures. Accordingly, mid-crustal RHP values were adjusted to produce reasonable geotherms consistent with stratified continental crust models (Rudnick and Gao, 2003; Hacker et al., 2015).

The range spanned by the seven scenarios tested for each terrane should be interpreted as an expression of structural uncertainty in the assumed crustal stratification, not as statistical confidence bounds on a fitted solution. The scenarios are constrained by surface radiogenic heat production measurements, geological interpretations, thermochronologically derived exhumation rates and literature-derived crustal structures; however, they are not calibrated against any direct thermal observation, because no surface heat-flow determinations and no deep borehole temperature data exist within the study area. The models are therefore not validated in the formal sense, since validation requires comparison against independent observations of the modelled quantity, and no such observations exist. They are instead assessed for internal consistency and geological plausibility: the requirement that Moho and mid-crustal temperatures remain compatible with petrological and geophysical constraints provides the principal external check, and is the criterion by which the preferred scenarios were selected. The modelled temperatures and heat flow values should accordingly be read as bracketed scenarios rather than deterministic predictions. In this regard, systematic xenolith studies of the exposed intrusions would provide a valuable independent constraint on deep crustal composition and RHP, and would represent an important target for future work.

In active collisional settings, additional dynamic and transient processes can influence crustal temperatures and heat flow, including exhumation, shear heating, partial melting and melt migration, crustal thickening, and paleoclimate effects. Shear heating is significant in fault zones during orogenic evolution (Burg and Gerya, 2005; Wang et al., 2013; Ai et al., 2021), but its crustal-scale contribution is negligible compared to radiogenic heat (Fagereng and Biggs, 2019). Melt migration from mid-crustal anatexis can advect heat to the upper crust, but, in the present case, its effect is spatially limited and neglected. Crustal thickening increases the total radiogenic content but is also excluded due to its long timescale relative to the model duration.

Lateral and vertical lithological variations, along with deformation intensity, strongly influence rock properties and can significantly affect local heat flow (Abdulagatova et al., 2020; Weinert et al., 2020). Thermophysical parameters used in the 2D model, derived from outcrop analogues, do not fully represent deep-crustal conditions affected by temperature, pressure, and fluids. Fluid circulation and rock permeability are not included; cold meteoric water percolation can cause hydrothermal cooling and locally reduce conductive geotherms (Cao et al., 2019). Conversely, in hot spring areas, upward advection of heat by fluids can elevate near-surface temperatures and disturb conductive heat-flow (Bächler et al., 2003; Koltzer et al., 2019). Consequently, regions with high deformation and fracture-related permeability are most susceptible to advective heat transport.

Simulating hydrothermal circulation would require local hydrogeological and structural constraints, such as fault permeability, fracture-network connectivity, and groundwater recharge. These parameters are not available and lie beyond the scope of a regional model. The conductive temperature field presented here therefore provides a background thermal state: it suggests that sufficient heat is available at depth to sustain hydrothermal systems, but not the specific flow pathways by which that heat reaches the surface. The link between the modelled temperatures and the observed springs is accordingly indirect.