All four components of the radiative heat fluxes at the surface – i.e. incoming and outgoing short-wave (S) and long-wave (L) radiation – were measured directly by the PERMOS micro-meteorological station with two back-to-back pairs of Kipp & Zonen CM3 pyranometers (0.3–3 µm) and CG3 pyrgeometers (5–50 µm) . We applied the same radiation corrections as in : (1) instrumental corrections of both pyranometers with nighttime short-wave radiation measurements (pyranometer offset correction), (2) correction for snow cover of the upward-looking pyranometer (S↓=S↑/0.86 when S↑>S↓), (3)  correction of the incoming long-wave radiation for interference with solar radiation, and (4) rejection of incoming short-wave radiation at low sun angles (if albedo exceeds unity).

We estimated the sensible turbulent flux QH using the eddy-covariance method, based on the covariance of vertical wind speed fluctuations w′ and temperature fluctuations T′, measured by a CSI CSAT and averaged over 30 min (; see Sect. ): 1QHeddy=ρaCpw′T′‾, where (ρaCp) refers to the volumetric heat capacity. We calculated buoyancy fluxes from the eddy raw data (10 Hz sampling rate) using Campbell Scientific's software EasyFlux™ and following the conventional pre-processing chain . Processing steps were trend and outlier removal, despiking , coordinate rotation using the double rotation method , and spectral corrections .

Due to the lack of fast-response moisture measurements (no high-frequency gas analyser), the eddy-covariance method yielded the sonic buoyancy flux w′Tsnc′‾, which, for |QH|>|QLE|, is close to but not equal to the sensible heat flux w′T′‾ . We assess the discrepancy using the Bowen ratio method modified by based on (“SND correction”): 2w′T′‾=w′Tsnc′‾1+0.51T‾aCpLvBo-1, where Tsnc is the sonic temperature, T‾a is the air temperature (averaged over 30 min), and Bo refers to the Bowen ratio (Sect. ). Cp and Lv are constants defined in Sect. .

We measured snow height with a sonic distance sensor (Table ) on a rock glacier ridge in addition to the nearby PERMOS snow height measurements. Raw measurements were compensated for variations in the speed of sound with air temperature, following the manufacturer's guidelines .

We used the semi-empirical ΔSNOW model to convert the measured snow height to snow water equivalent (SWE). This parsimonious model requires snow height and its temporal changes as the sole input. Calibration data used to develop their model were gathered in the Swiss and Austrian Alps in climatic regions similar to the Engadine.

We measured liquid precipitation with an unheated tipping-bucket rain gauge (Table ) mounted on a rock glacier ridge.

Snow or sleet (a mix of snow and rain) can fall in any month of the year. The distinction from rain is important because of the latent heat of melting Lm that ice carries and by far exceeds the sensible heat of rainwater (cwΔT). In the comparatively dry climate of the Engadine, wet-bulb temperature can be a better discriminator than dry-bulb temperature . Wet-bulb temperature Twb [K] is defined in : 3e-e∗(Twb)=-γ(T-Twb)[Pa], where γ=(cpP)/(0.622Lv) [PaK-1] is the psychrometric “constant”. Equation () is solved iteratively. Based on hourly images from the automatic time-lapse camera, we defined a conservative wet-bulb temperature threshold of 2 °C that separates snow or sleet (Twb<2°C) from rain.

The in-cavity net long-wave radiation QCGR3rad was calculated from the upwards Lal↑ and downwards Lal↓ long-wave radiation components measured with a back-to-back pyrgeometer pair installed in the instrumented cavity at a depth of 1.55 m beneath ground level (Table ): 4QCGR3rad=(Lal↑-Lal↓), which is not to be confused with the long-wave radiation L measured at 2 m a.g.l. (Sect. ).

We corrected raw outputs Lraw↑/↓ from the two pyrgeometers in the instrumented cavity by accounting for the long-wave radiation emitted by the instruments themselves : 5Lal↑/↓=Lraw↑/↓+σTCGR34, where TCGR3 refers to the pyrgeometer housing temperature. Large (>0.5 °C) or rapid changes in housing temperature differences between the two back-to-back mounted pyrgeometers indicated dust or water deposition on the upward-facing pyrgeometer window. Such disturbed measurements appeared in the high-resolution (10 min) data but did not significantly affect the daily net long-wave radiation balance in the sheltered cavity.

We refer to the sub-surface “wind” in the cavity as “airflow” to differentiate it from the atmospheric wind. We used the Hukseflux WS01/TP01 sensor to perform airflow speed measurements in the AL cavities. This sensor included a heated foil that measures a cooling rate expressed as a convective heat transfer coefficient hWS [Wm-2K-1] related to airflow speed . We did not convert hWS to airflow speed but process this variable as a semi-quantitative indicator assuming u∝hWS, sufficient for our purpose. WS01/TP01 do not resolve the direction of the airflow (hence the term “speed” instead of “velocity”). Deposition and evaporation of liquid water can disturb measurements (hWS increase), as revealed by wrapping the heated foils in moist tissues. WS01 measurements during precipitation events were filtered out. Repeated zero-point checks were performed throughout the snow-free season by enclosing the heated foil in small, dry plastic bags for a few hours, ensuring stagnant conditions with zero airflow speed. Neither drift nor temperature dependency beyond measurement uncertainty was detected.

The net radiation Q∗ is the sum of the measured and corrected radiation components (Sect. ): 7Q∗=S↓+S↑︸S∗+L↓+L↑︸L∗, where S↑ is the outgoing short-wave radiation, S↓ is the incoming short-wave radiation, and net short-wave radiation S∗ is the sum of both; correspondingly, L↑ represents the outgoing long-wave radiation, L↓ is the incoming long-wave radiation, and net long-wave radiation L∗ is the sum of both.

We estimated the turbulent flux using three different methods: (1) the Bowen energy balance method , (2) the bulk aerodynamic method , and (3) directly with the eddy-covariance method from CSAT measurements (in the lack of a fast-response vapour analyser, only the sensible turbulent flux QH is estimated; Sect. ).

The Bowen ratio Bo is defined as the ratio of sensible to latent heat flux and reflects the partitioning of the turbulent fluxes into sensible and latent components: 8Bo:=QHQLE=KhKwCpΔTLvΔq≈Cp(Ta-Ts)Lv(qa-qs), where Kh represents the eddy diffusivities for sensible heat and Kw is water vapour. Invoking the similarity principle and assuming Kh≈Kw , the Bowen ratio can be calculated from the isobaric specific heat capacity of air (Cp=Cpd(1+0.84q) with Cpd=1005 Jkg-1K-1) , the latent heat of vaporisation Lv (if Ts≥0 °C) or sublimation Ls (if Ts<0 °C) (2.48×106 Jkg-1, 2.83×106 Jkg-1), and the gradients of temperature ΔT and specific humidity Δq (specified below). The sensible and latent turbulent fluxes are then expressed as a function of the available energy Q∗+QM+QG/S (QP considered negligible; Sect. ): 9QHBo=Q∗+QM+QG/S1+1/Bo,QLEBo=Q∗+QM+QG/S1+Bo.

The ground and snow surface temperature Ts [°C] was calculated from the measured long-wave radiation components L and the Stefan–Boltzmann law via 10εσTs4=L↑-(1-ε)L↓, where ε is the surface emissivity; σ represents the Stefan–Boltzmann constant (σ=5.670×10-8 Wm-2K-4); and L↑ and L↓ represent the measured outgoing and incoming long-wave radiation, respectively. We took the emissivity value of ε=0.96 from .

The surface specific humidity qs [gg-1] is either the specific humidity of the air in the near-surface coarse blocky AL qsa (measured) or the saturated snow surface qss∗(Ts,Pa), depending on whether or not the snow cover is thick enough to suppress convective air exchange between the AL and the atmosphere (Sect. ). 11qs=qsa,forhS≤SEBh^Scrit(snow-free or open snow cover)qss∗,forhS>SEBh^Scrit(decoupling snow cover) The critical snow height SEBh^Scrit is reached when the AL is so strongly decoupled from the atmosphere that the convective fluxes across the snow cover are no longer detectable by our SEB estimations. We determine the snow height SEBh^Scrit using the AL air temperature, specific humidity, and airflow speeds (Sect. ). The specific humidity of the air qa was calculated from the measured air temperature Ta and relative humidity at the measurement level zm of 2 m a.g.l.

In the bulk parameterisation, sensible QHbulk and latent QLEbulk turbulent fluxes are driven by the gradients of temperature ΔT=Ta-Ts [K, °C] and specific humidity Δq=qa-qs [gg-1], respectively, and horizontal wind speed u [ms-1]: 12QHbulk=ρaCpu(Ta-Ts)Cbt,13QLEbulk=ρaL{v,s}u(qa-qs)Cbt, where ρa [kgm-3] represents air density. The flux–gradient relationship is modified by atmospheric stability, accounting for enhanced turbulent fluxes in an unstable atmosphere and suppressed turbulent fluxes in a stable atmosphere, by means of the bulk exchange factors Cbt (bulk turbulent heat and vapour transfer coefficients) [unitless, –]. Different formulations of the stability functions exist. Here, we use the modified scheme , motivated by its use in the GEOtop model, a distributed hydrological model designed for complex terrain , and other studies (e.g. ). We compare the modified scheme with the iterative Monin–Obukhov scheme and the widely used Businger–Dyer scheme . The latter has been used in the previous SEB studies on Murtèl by and and in many SEB estimates on debris-covered glaciers , despite discrepancies (overestimated fluxes) at strongly unstable atmosphere noted by . The issue with the Businger–Dyer parameterisation is that near-surface wind speeds on topographically sheltered rough terrain tend to be lower for a given atmospheric stability compared with the conditions under which the empirical Businger–Dyer parameterisation was originally developed (flatland in Kansas, USA; ; ). This approach is therefore problematic in complex terrain, including our study site. Finally, we test the parameterisation without stability correction (“bulk c0”), which corresponds to the special case of a neutral atmosphere. Detailed explanations of the parameterisation schemes of the turbulent heat transfers are described in the appendix (Appendix ).

Accurate estimates of turbulent fluxes rely on representative values for the roughness lengths for momentum z0m, heat z0h, and moisture z0q . We calculated the roughness length for momentum z0m from the eddy-covariance data following and : 14z0m=(zu-d)exp⁡-k∗u‾hu∗-ψmzu-dL, where u‾h [ms-1] represents the mean horizontal wind speed, u∗ [ms-1] is the friction velocity, k∗ [0.40] is the von Kármán constant, zu [m] is the measurement height of wind speed, L [m] is the Obukhov length, ψ is the integrated stability function (Appendix ), and d is the zero-plane displacement height. We set the (unknown) zero-plane displacement height d to zero . The sensor height zu(t) is the variable distance above the ground or snow surface: zu(t):=zCSAT-hS(t). The recommended filtering is to use only u‾h>3 ms-1 under near-neutral conditions |zu/L|<0.05 . We compared our eddy-covariance-derived roughness length for momentum z0m with the aerodynamically derived values reported in . For simplicity, the scalar lengths for heat z0h and humidity z0q are often considered equal to the momentum roughness length z0m or 1–3 orders of magnitudes smaller . Here, we could not independently estimate the scalar roughness lengths z0h and z0q due to the lack of humidity-corrected sonic temperature or high-frequency humidity measurements. We assumed an equal roughness length for heat and moisture, z0h=z0q, and used the unknown ratio R^z0:=z0h/z0m=z0q/z0m as a calibration parameter.

Finally, we compared the sensible turbulent flux with the katabatic model of specifically developed for conditions of katabatic or nocturnal drainage winds. This is different from all the above-mentioned parameterisations as this predicts a quadratic rather than near-linear relation between QH and the driving temperature difference ΔT. Since we lacked vertical profile observations of potential temperature required for this parameterisation, we could not test this bulk method. Instead, we checked the validity of the quadratic QH–ΔT relation on Murtèl (cf. ) and show wind profile measurements collected by in the years 1997–2000 (Appendix ).

The snowmelt heat flux QM is the heat flux consumed by the melting snowpack. We estimated QM from the daily change in SWE [kgm-2] derived from the snow height data hS (sonic ranger data) with the semi-empirical parsimonious ΔSNOW model : 15QM≈LmΔ(SWE)Δt, if Tb=0°C, where Lm [336 kJkg-1] represents the latent heat of fusion for ice. We considered runoff-generating snowmelt to occur in spring when temperature at the base of the snowpack Tb reaches the melting point and ignored melting–refreezing events within the snowpack.

The snow heat flux QS is the conductive heat flux across the single-layer snowpack. We calculated Q̃S in the lowermost 25 cm of the snowpack, ignoring transient effects : 16Q̃S=-kSdTdz≈-kSTs-TbhS≈-kST25-T0Δz, if hS>30cmand0otherwise, where dT/dz is a linearised temperature gradient in the lowermost 25 cm of the snowpack. To ensure that both thermistors are snow covered, the minimum snow height hS required is 30 cm. We related the snow thermal conductivity kS to the snow density via the empirical equation kS:=kS(ρS)=2.93(10-6ρS2+10-2) developed for Murtèl , where we estimated the bulk density of the snowpack with the ΔSNOW model via ρ‾S=SWE/hS .

Q̃S was calculated near the base of the snowpack instead of at the snow reference surface to which Eq. () refers. With this approach, we used the more stable snow density and thermal conductivity near the snowpack base, which is less affected by compaction than the near-surface layers that receive fresh snow. The heat flux Q̃S was extrapolated to the snow surface by adding the sensible heat storage changes in the snowpack ΔHS above zS=25 cm (“calorimetric correction”) . We estimated the changes in cold content ΔHS as 17ΔHS≈cS(mS/A)ΔTS/Δt≈cSSWE〈ΔTS〉/Δt, where (mS/A) represents the mass of the snowpack per area, i.e. the SWE [kgm-2]; cS [2.050kJkg-1K-1] is the specific heat capacity of snow or ice at 0°C; and 〈ΔTS〉 [°C] represents the layer-averaged snow temperature changes. The calorimetrically corrected snow heat flux QS was then calculated as 18QS=Q̃S+ΔHS.

The rainfall heat flux QPr was estimated via 19QPr=Cwr(TP-Ts), where Cw=ρwcw (4.18 MJm-3K-1) is the water volumetric heat capacity and r [m3m-2s-1] is the rainfall rate intercepted at the surface. Precipitation temperature TP was approximated using the wet-bulb temperature Twb, calculated from air temperature and relative humidity (Eq. ). We assumed precipitation in the form of rain if Twb≥2 °C. Water contributions from upslope flowing onto the rock glacier and liquid precipitation falling into the snowpack were not accounted for.

The heat flux QPs due to rapid melting of sleet or shallow summertime snow (Twb<2 °C, hS<10 cm) was roughly estimated as 20QPs∼Lmρwrs, where rs [m3m-2s-1] represents the amount of solid precipitation that melts in the pluviometer per time period Δt [s]. Our measurement setup was not designed to accurately record the precipitation rate during mixed rain- and snowfall or the fraction of ice crystals and liquid water in the total precipitation. The heat flux QPs is intended as an order-of-magnitude estimate.

Analogously to the calorimetrically corrected snow heat flux QS, the ground heat flux QG is the sum of the measured net long-wave radiation in the instrumented cavity QCGR3rad (Eq. ) and the sensible heat storage changes ΔHalθ of the virtual layer between the ground surface and the depth of the pyrgeometer: 21QG=QCGR3rad+ΔHalθ.

Rock mass subject to changing temperatures constitutes a heat source or sink . The sensible heat storage change in the (dry) blocks ΔHalθ is proportional to the rate of change in rock temperature, assuming a constant volumetric heat capacity ∂{(1-ϕal)ρrcr}/∂t=0. In the lack of rock temperatures over the 2-year time period analysed in this work, we use in-cavity air temperatures T‾a averaged over a thermal adjustment timescale Δtr (1 d) as a surrogate (Sect. ; Appendix ). The rate of sensible heat storage and release of the blocks was then approximately calculated as 22ΔHalθ=∫-zs0∂∂t{(1-ϕal)ρrcrTr(z)}dz≈(1-ϕal)〈ρrcr〉Δtr∑i{〈T‾a(zi,t+Δtr)〉-〈T‾a(zi,t)〉}Δzi, where zs=155 cm is the distance from the pyrgeometer pair to the ground surface (reference level); ρr is the rock density (2690 kgm-3) (Corvatsch granodiorite; ); cr is the specific heat capacity (790 Jkg-1K-1); ϕal=0.4 is the AL porosity; and Tr(z) and Ta(z) are the vertical rock and in-cavity air temperature profile [°C], respectively. In the discretised formulation, temperatures 〈T‾(zi)〉 are layer-wise averages in the ith layer with thickness Δzi (denoted by “〈⋅〉”) derived from the thermistor string TK1/1 and the radiometric surface temperature Ts.

Short- and long-wave radiation are by far the largest SEB components (Fig. a). Consequently, the well-known effect of the snow cover on albedo is the single largest control of net radiation and hence of the entire SEB (Fig. b).

In mid-winter, the meso-scale relief controls the solar radiation budget. The shaded north-facing Murtèl cirque receives no direct insolation from November to February. Depending on cloud cover and amount of incoming diffuse or terrain-reflected short-wave radiation, the net radiation is negative and dominated by the long-wave radiation budget (L∗≪S∗<0). Net radiation Q∗ was less negative in the cloudy and precipitation-rich winter 2020–2021 (December–February; average: -19 Wm-2) than in the sunny and dry winter 2021–2022 (-40 Wm-2) due to greater incoming long-wave radiation L↓ emitted by the clouds (233 vs. 215 Wm-2). A case in point is the exceptionally warm and sunny November 2020, which received in total 33 Wm-2 less short- and long-wave radiation than November 2021. The monthly mean radiation deficits are up to -53 Wm-2, which, together with the steep snow-covered slopes, is a favourable setting for strong katabatic winds.

The Bowen ratio partitions the available energy from net radiation, the ground heat flux, and the snowmelt heat flux into sensible and latent turbulent fluxes. Throughout the seasons of both years, the heat fluxes were as follows (Fig. ): during wintertime, from November to March, without direct insolation in the shaded cirque, the energy loss of 20–50 Wm-2 due to long-wave emission and terrain shading was largely compensated by the sensible heat flux QH. Latent heat flux QLE at the cold snow surface was small, with mostly resublimation (moist winter 2020–2021) and sublimation fluxes (dry winter 2021–2022) within -6±10 Wm-2 (Fig. a). With the latent heat flux being small and in a variable direction, the computed Bowen ratio showed a large scatter (|Bo|>1; Fig. b). In the drier winter 2021–2022, QH compensated for the heat lost by sublimation in addition to the radiative heat loss. During spring, from April to May, before the snowmelt period, the magnitude of the fluxes remained similar but the direction reversed. In summer, after complete snow melt-out, from June/July to October, the turbulent fluxes were larger to compensate for the large radiation surplus. In the more rainy summer 2021, sensible and latent heat exports were similarly large (Bo=1.25±2.71), whereas in the drier summer 2022, heat export was more dominated by sensible flux (Bo=1.48±2.47). Towards early autumn, with decreasing temperatures, the sensible flux lost importance relative to the latent flux (Bo decreased and approached zero). In September/October, with mixed precipitation (sleet) and first snow falls at still warm conditions near 0 °C, the latent heat export by sublimation from the snow cover intermittently dominated over the sensible heat uptake (Bo between 0 and -1). With further cooling, the moisture supply became limited again, and sensible heat uptake took over to offset the increasing radiation deficit (Bo<-1).

We compare the estimates of turbulent fluxes (monthly averages) with the net radiation (Fig. ) and among each other (Fig. ). In the snow-free summer months, the bulk fluxes tend to slightly underestimate the fluxes, while the Businger–Dyer fluxes (cB&D) clearly overestimate them. The measured eddy-covariance flux (buoyancy flux) falls short of closing the SEB, roughly by 50 %–75 % depending on how large the (unmeasured) eddy latent flux would be. The imbalance (“missing” flux to close the SEB; ) is, in any case, substantial.

The latent heat of the melting snowpack QM largely consumes the available net radiation Q∗ in the respective snowmelt months; the snowmelt heat flux is roughly as large as typical summer sensible turbulent heat fluxes QH (Fig. ).

The sensible rain heat flux QPr exerts a negligible cooling effect of ≤-2 Wm-2 at the daily timescale (not shown). Short but intense rainfall (thunderstorms) with fluxes up to -100 Wm-2 in 10 min is averaged out because such high-precipitation events are short. The heat fluxes QPs arising from mixed precipitation or a shallow snow cover that melts within hours (“summer snow”; 10–30 Wm-2 daily average) are similar to QG and not negligible. Such events typically occur in early autumn (September–October) but also occurred throughout the wet and cool summer 2021 (June–October). Their timing often coincides with episodes of rapid ground cooling (Fig. ).

Snow heat flux QS is in the range of 2 to -4 Wm-2 and ground heat flux QG in the range of 30 to -20 Wm-2 at the daily timescale (Fig. ). The snowpack is a thermal insulator. Within-snowpack conductive fluxes are 2–3 orders of magnitude smaller than typical SEB components. The conductive snow heat flux is negligible compared with the overall SEB and across-snowpack convective fluxes (snow funnels; Figs. , ). The simplifications from the single-layer snowpack that ignore vertically varying snow density do not detract from this finding.

Heat storage changes in the snowpack (Fig. a) and uppermost coarse blocky AL (Fig. b) are asymmetric in opposite directions. In the snowpack, downwards heat transfer or warming (storage gain) is more intense (larger maxima) but less frequent, likely due to non-conductive fluxes from refreezing meltwater (synchronised with warm spells Ta>0 °C or rapid warming to the zero curtain). In the near-surface AL, upwards heat transfer or cooling (storage loss) is more intense but less frequent. Different weather conditions in the two summers analysed are reflected by QG: the passage of cold fronts in summer 2021 led to more frequent convective cooling, often accompanied and enhanced by sleet; conversely, ground warming is pronounced during dry spells in summer 2022. Similarly for the two winters, virtually no storage changes (QG≈0) occurred in the snow-rich winter 2020–2021, whereas temperature and sensible heat storage fluctuations continued in the snow-poor winter 2021–2022, albeit strongly attenuated compared with those in the snow-free season (Fig. ; Sect. ). The non-zero QG in winter 2021–2022 beneath the thin snow cover reflects convective heat exchange across the snow funnels since the conductive snow heat flux QS is too small to account for QG.

Monthly SEBs are closed within the calculation uncertainties of ≤20 Wm-2 (Fig. , blue bars; Table ). This represents a substantial improvement compared with and and is due to the novel sensor array used in the present work. The largest SEB imbalances (deviation from closure) occur during the transition between seasons, when ground thermal conditions linger around 0 °C, during extreme meteorological conditions, and in mid-winter (December–March). These are the snowmelt months (June 2021, May 2022), early autumn (September), and the July 2022 heat wave and accompanying dry spell that strongly impacted the ground thermal and moisture regime (Figs. –). Larger deviations that occurred in the snow-rich mid-winter 2020–2021 are reduced by a “katabatic wind correction” for the variable anemometer height above the snow surface (Sect. ).

Uncertainties arise from terrain or snow cover variability across the rock glacier, spatially variable properties of the coarse blocky AL (e.g. emissivity, porosity, intrinsic permeability), and instrumental measurement errors, among other factors. We assessed the impact of the largest sources of parameter uncertainty of the heat fluxes (Table ; sensor accuracy from ). We estimate our overall accuracy as ∼20 Wm-2 at the daily timescale due to uncertainties in QG (uncertainties in sensible heat storage changes ΔHalθ) and QH (uncertainty in surface temperature Ts propagated from the emissivity ε; Eq. ). These uncertainties play an important role in this study. The SEB uncertainties determine which processes are included in the SEB estimates and which are not, complementing measurement-driven criteria. We consider insignificant (i.e. within the uncertainty) the processes with associated fluxes smaller than our 20 Wm-2 uncertainty threshold. The processes considered insignificant in the context of the SEB are not necessarily insignificant at depth. In fact, in the AL, all daily-averaged sub-surface heat fluxes are within 20 Wm-2 (Fig. b). We apply this criterion on the nocturnal Balch ventilation (Sect. ) and the decoupling snow height SEBh^Scrit (Sect. ).

Insolation differences due to the local shading effect of the coarse terrain surface, the terrain slope effects, or instrumental effects (cosine response) might lead to differences of up to 30 Wm-2 (daily average) in summer (Table ) . Uncertainties in the outgoing long-wave radiation L↑ of ±10 % corresponding to a radiometric surface temperature difference of 7 °C might arise from patchy snow cover. The lateral advective heat transport altering the boundary layer characteristics could explain the large SEB imbalance (Fig. ) of roughly 100–150 Wm-2 during the melt-out phase. Smaller but still significant Ts differences of 2 °C can occur due to local shading (micro-topography) or uncertainty in emissivity (e.g. variable snow emissivity; ) and lead to considerable QH uncertainties of up to 50 Wm-2. Surface temperature differences from different emissivity values (ε=1 instead of 0.96 from ; Eq. ) are within -1.2 and +0.25 °C. The deviations tend to be largest on hot clear-sky days with little incoming long-wave radiation L↓ (Eq. ) and hence might translate into considerable uncertainties in QH precisely when these are largest.

Temperature and specific humidity uncertainties driving bulk fluxes (Eqs. , –) lead to considerable QH and QLE uncertainties. The parameterisation is moderately sensitive to the spatially variable wind field which might arise from the micro-topography, for example wind sheltering in the furrows. The momentum roughness length z0m is varied by a factor of 100.5 (between 0.09 and 0.45 m), which reflects the standard deviation of the measurements (Fig. ). The roughness lengths are perhaps among the most critical parameters for the estimation of turbulent fluxes when using the bulk approach (besides the emissivity ε): sensitive yet hard to constrain on rough and complex mountainous terrain . Equivalence between momentum z0m and scalar roughness lengths for heat z0h or moisture z0q would lead to prohibitively large deviations and can be excluded. As the rough terrain turns sensor height above the surface into a somewhat vague parameter, we vary it by a typical block edge length of 0.5 m. The arising QH deviations of 10–20 Wm-2 are similar to other parameter uncertainties. Air temperature and specific humidity measurements from the PERMOS and PERMA-XT stations, located within 50 m, differ by 0.6±1.0 °C and 0.004±0.24 gkg-1, respectively. The calculated fluxes are similar: QH is within ±5 Wm-2 and QLE within 2 Wm-2. Maximum deviations are temporarily up to 10–15 Wm-2 in winter and during the snowmelt period. The PERMOS and PERMA-XT station data yield flux estimates indistinguishable within their uncertainty.

Ground heat flux QG primarily reflects the rate of temperature change (RTC) of the near-surface AL and is weakly sensitive to the pyrgeometer flux measurements QCGR3rad. The reason behind is that QG (Eq. ) is dominated by the sensible heat storage changes ΔHalθ of the 1.5 m thick blocky layer above the long-wave radiation measurement . Consequently, rather large uncertainties in QG (20 Wm-2) come from uncertainties in the storage changes δ(ΔHalθ) from two factors, namely the porosity ϕal and the thermal adjustment time Δtr (time lags). First, a high porosity limits the heat storage capacity that scales with 1-ϕal. We tested a plausible range of δϕal±0.2 and obtained uncertainties up to 20 Wm-2. Porosity might laterally vary closest to the surface, precisely where daily temperature amplitudes are largest. Second, the assumption of similar air and rock temperature profiles T‾a(z)≈Tr(z) – local thermal equilibrium (LTE) assumption ) – becomes problematic in the roof of the ventilated cavity, where conditions are highly transient (convective heat transfer and effect of insolation). The entire rock mass might not adjust to the rapid temperature fluctuations of +2 to -3 °C from day to day. The chosen thermal relaxation time Δtr of 1 d is a minimum duration (Appendix ). We assess the influence of longer adjustment times Δtr by comparing 1 d with 5 d running averages. The differences are similar to the uncertainties related to porosity. We conclude that, due to the large and rapid heat turnover in the ventilated near-surface coarse blocky AL under highly transient conditions, the ground heat flux QG is arguably the least-constrained flux. This might not be a surprising finding on a landform that does not present a clearly defined surface.

Finally, we let SEBh^Scrit=80 cm to simulate an open snow cover in early–mid-winter 2020–2021 (October–February; Fig. ) and throughout the snow-poor 2021–2022. Due to the large humidity differences across the snow cover (Fig. ), this resulted in a massive (up to 102 fold) increase in QLE. The closing of the snow cover is a key control factor for the SEB and ground thermal regimes (Eq. ; Sect. ) .

In the sensitivity analysis (Table ), input parameters and meteorological variables are varied independently from each other and based on likely maximum measurement errors. However, the meso-scale relief and local sub-surface processes like ventilation might add some systematic biases that exceed instrumental errors. We will explore parameterisation uncertainties (stability corrections) in Sect.  and uncertainties in the meteorological input variables in Sect. .

The interaction of the steep snow-covered wintertime terrain with a negative radiation balance induces downslope katabatic winds that govern the near-surface wind field. This, in turn, affects the calculation of the bulk turbulent fluxes that require a representative wind speed as input. Our initially calculated wintertime turbulent fluxes were “too large” by 10–35 Wm-2 on monthly average (Fig. ), especially during the overcast snow-rich winter 2020–2021, while the radiative cooling and the forcing temperature deficit ΔT were weaker than in the snow-poor winter 2021–2022 (Fig. ; Appendix ; Eqs. –). This result suggests that measured wind speeds were off in relation to the snow height and not to the temperature deficit. In fact, wind tower measurements on Murtèl performed by showed strong and persistent katabatic winds in winter, with a wind speed maximum a few metres above the surface weakly correlated to snow height (Appendix ; see Fig. c). The growing snow cover “shifted” the high-velocity region of the low-level katabatic jet to the level of the wind sensor, causing an apparent wind speed increase. We compensate the wind speed us measured at variable height above the snow cover (zs-hS) for the snow height with a simple power-law relation: 23u=us(zu/(zs-hS))-m^, where m^ is the exponent. Note that the usual correction of the sensor height above the variable snow surface further increased rather than decreased the turbulent fluxes. Since our measurements cannot resolve the near-surface wind speed profile of the low-level katabatic jet, the intention of this ad hoc “katabatic wind correction” is not to accurately describe the wintertime wind profile but rather to pragmatically render our input data amenable to the flux–gradient parameterisations based on the Monin–Obukhov theory. We use m^ as a calibration parameter (denoted by the circumflex). A literature value of m^=0.6 for stable conditions reduced the wind speed sufficiently to yield turbulent fluxes that close the SEB within our uncertainty threshold (Fig. ). Although the Monin–Obukhov theory is not strictly valid under such conditions , our consistent flux estimates based on a scaled wind speed are in line with and , who argue that the bulk method still provides reasonable flux estimates when measured close to the surface, as was the case here (measurements within 2 m above the variable snow cover). This illustrates the importance of accurate wind speeds for the calculation of turbulent fluxes. We suggest an alternative solution in Sect. .

From the comparison of the measured eddy-covariance fluxes and the calculated Bowen and bulk fluxes (daily averages), we reach the following conclusions.

Overall, the modified parameterisation (cL′) seems the best choice to parameterise the turbulent fluxes on Murtèl for three reasons. First, as expected, the modified cL′ bulk fluxes were near-identical to the iteratively calculated Monin–Obukhov fluxes (Fig. d; r=0.996) but at less computational cost. Second, neither filtering nor post-processing was necessary because virtually all estimates met the quality criteria. Third, these show the least deviations from the Bowen fluxes (Fig. e), which we consider the benchmark as the Bowen SEB is closed by design. The Bowen fluxes offer reasonable daily estimates based on minimal measurement requirements, namely temperature and specific humidity profiles. Wind speed is not required. The minimum time resolution is daily scale. Hourly values either are numerically unstable (small ΔT/Δq, especially over a “warm” spring–early-summer snow cover) or do not account for the systematic diurnal wind speed variations. In summer, for example, hourly sensible Bowen fluxes are overestimated during the calm nights and compensated by excessive daytime fluxes.

The eddy-covariance flux systematically underestimates the sensible turbulent flux by a factor of ∼2, leading to a large imbalance (underclosure) in the eddy SEB. The error from the lacking SND correction (Sect. ) cannot account for the eddy SEB imbalance. The ratio between the eddy sensible flux and the (sonic) buoyancy flux is between 0.93 and 1.05 (10 % and 90 % quantile, respectively; Eq. ) on a daily average. Since both the air and the ground surface are most often far from saturation, the eddy sensible flux error due to the lacking high-frequency gas analyser is less than 10 % on 617 out of 692 d (89.2 %) with valid Bowen ratios. Larger deviations occur on the remaining 75 d with |QLE|>|QH| (Bowen ratio within -0.6 and 0.5), typically during snowmelt periods (“warm” saturated snow surface), cloudy and rainy summer days (more frequently in summer 2021 than 2022), or during the first snowfall in autumn. Applying the SND correction showed little added value. Nonetheless, the summertime eddy-covariance fluxes correlate reasonably well with the Bowen sensible turbulent fluxes (Fig. a) and the modified bulk fluxes (Fig. b) or the driving temperature gradient (Fig. a), and therefore random instrumental or data processing errors represent an unlikely explanation. We hypothesise that some hidden systematic reason causes the flux underestimation and the large eddy SEB imbalance, e.g. secondary circulation of the anabatic afternoon winds.

The empirical Businger–Dyer (cB&D) formulation, developed over flat terrain, overestimates the turbulent fluxes and deviates more strongly for larger negative fluxes (“banana” shape; Fig. f), despite extensive filtering. Even the bulk parameterisation without stability correction (c0) outperformed cB&D for the summer fluxes under unstable atmospheric conditions (Fig. c). This finding agrees well with on the Lirung debris-covered glacier that shares many topo-climatic features with the Murtèl rock glacier (unstable atmosphere over a strongly heated debris surface with anabatic valley winds).

Finally, we describe the relation between the wintertime sensible Bowen fluxes with the driving temperature gradient ΔT (“surface temperature deficit”) as quadratic (Fig. b). A quadratic functional QH–ΔT relation is unique to the katabatic model of . Note that the Bowen fluxes are independent of wind speed (Eq. ). Together with the ad hoc “katabatic correction”, our data show that the turbulent fluxes in a snow-covered cirque with katabatic winds might be better parameterised by a katabatic model than by the common Monin–Obukhov bulk method, as also found by on a sloped glacier surface and discussed by in the context of atmospheric boundary layer modelling.

The snow cover controls the coupling between the coarse blocky AL and the atmosphere. The turbulent fluxes draw heat and moisture from the AL as long as the snow cover is “open” via snow funnels, a phenomenon widely observed on coarse blocky landforms . A thickening snow cover gradually suppresses the vertical convective coupling between the AL and the atmosphere, as more snow funnels close (sketched by the schematic envelopes in Fig. ). A snow height of ∼10 cm begins to decouple the coarse blocky AL from the atmosphere (“semi-closed” in Fig. ), but much more snow (∼70 cm) is necessary to achieve the insulating effect of the snow cover on the ground thermal regime (“closed/insulating”). Then, the snow cover is thick enough to suppress convective air exchange, rapid sub-daily fluctuations are strongly attenuated, and large temperature and moisture gradients to the outside air build up , indicating that the AL–atmosphere coupling is weak. Such a high value is typical for a terrain as rough and blocky as on Murtèl, agreeing with, for example, and . The insulating effect of a thick snow cover was already observed decades ago and used to indirectly map the permafrost distribution using the bottom temperature of snow cover (BTS) method .

As regards the SEB, a snow cover as thin as SEBh^Scrit=20 cm causes a decoupling that is strong enough for our SEB to be significant. This is shown in the snow-poor winter 2021–2022 with snow depths between 40 and 70 cm, when the ground heat flux QG kept fluctuating but remained below the 20 Wm-2 uncertainty threshold (Table ). The convective flux across the snow cover cannot deviate strongly from QG because it was the single largest heat flux to supply/extract the heat for the AL sensible heat storage changes (small conductive snow heat flux, QS±2 Wm-2, Ts<0 °C, and no latent effects from snowmelt in that period). We take this threshold as the critical snow thickness (SEBh^Scrit; Eq. ). We emphasise that the 20 cm threshold is an operational definition for the “decoupled” snow cover in the context of our SEB and its uncertainties.

From Sect.  it follows that for the decoupling or insulating snow cover (hS≥SEBh^Scrit), the reference surface coincides with the snow surface. The radiometric surface temperature represents the surface temperature Ts (Eqs. , ). Whenever the snow cover is open or under snow-free conditions (hS<SEBh^Scrit), the radiometric surface temperature is the input meteorological variable Ts (Eq. ). The surface is where the solar radiation is intercepted and transformed to thermal energy, hence the main source of sensible heat for QH. The large gradient (Ta-Ts)/zT drives the sensible turbulent flux QH (Eqs. , ; Fig. a) and the wind-forced ventilation (Fig. b) . This is shown by the measured eddy-covariance flux (Fig. c) that largely follows the difference between air and radiometric surface temperatures (Fig. a, red area), as do the local (anabatic) wind in the atmosphere above the surface and the airflow speed in the near-surface AL (Fig. b). During daytime with strong heating on the low-albedo surface, the blocky surface exceeds air temperatures by >15 °C.

However, we found evidence of near-surface ventilation that cannot be parameterised by the 2 m air temperature and the radiometric ground surface temperature: nocturnal Balch ventilation, a nighttime cooling process owed to the interplay of the air permeability, and the thermal inertia (heat retention) of the coarse blocky AL that is most apparent during fair-weather summer days with clear nights. Upwards turbulent heat export is protracted into the evening hours, long after sunset (S∗; Fig. c), as shown by the measured in-cavity airflow speed (WS/3; Fig. b) and the eddy-covariance flux (Fig. c). Furthermore, the measured eddy-covariance flux remained upwards-directed in the early morning hours, when the terrain surface has radiatively cooled to air temperature Ts≈Ta or even during reversals Ts<Ta in clear-sky nights (that frequently occur in the Engadine: e.g. 24 out of 31 nights in August 2022). The calculated bulk fluxes driven by 2 m air temperature and radiometric ground surface temperature (rGST; red area in Fig. a; Eq. ) are then close to zero (or even downwards) and do not fully capture this nocturnal drawing of heat from the near-surface AL. The large thermal inertia of the rock mass and the protection from long-wave radiative cooling stabilise the sub-surface air temperatures and keep the air in the roof of the ventilated cavity during the nights warmer than that in the atmosphere outside and that on the ground surface (Fig. a; TK1/1≈TK6/1>Ta≈Ts by ∼4 °C). In the nighttime, the locally stably stratified air in the uppermost cavity roof is hence warmer and more unstable than that in the atmosphere (non-local static stability, ). The high permeability of the coarse blocky AL allows this air to escape upwards into the atmosphere or, equivalently, allows the colder outside air to sink into the cavity, thereby exporting heat (Balch ventilation; ). describes an occasional nocturnal sub-surface ventilation on Murtèl when Ts<Ta. Also interpreted nocturnal near-surface ventilation on the debris-covered Koxkar glacier (Xinjiang, China) from eddy-covariance and temperature data.

We assess this uncertainty by using the cavity roof temperature TK1/1 instead of the conventional 2 m air temperature as input Ta for the bulk fluxes (Eq. ) and compare the nighttime fluxes (when TK1/1 >Ts). Due to the low wind speeds at night, the estimated nocturnal Balch fluxes are small despite the appreciable temperature gradients (10–20 Wm-2), maximally 10 Wm-2 larger than the “conventional” bulk flux and within our SEB 20 Wm-2 uncertainty (Table ). Since the short-wave radiative forcing S∗ and the wind speed covary in phase, the “conventional” radiometric ground surface temperature and 2 m air temperature measurements are sufficiently adequate to parameterise the turbulent fluxes when sub-daily resolution is not required – especially considering that the chosen parameterisation (stability function) shows a much larger effect on the calculated turbulent flux (Fig. c; compare cL′ with cB&D). We reach a conclusion equivalent to that in Sect.  for the critical snow height: the nocturnal convective processes in summer are within the uncertainties of the daily-averaged SEB but might exert an important cooling effect on the sub-surface energy balance.

From Sect.  it follows that for the decoupling or insulating snow cover (hS≥SEBh^Scrit), the snow surface humidity at saturation qss∗ represents the surface specific humidity qs (Eqs. , ) . Whenever the snow cover is open or under snow-free conditions (hS<SEBh^Scrit), the humidity measurement in the cavity roof qsa is the input meteorological variable qs (Eq. ).

Perhaps contrary to the impression of a dry-looking blocky surface, the estimated summertime latent turbulent flux QLE is ∼ 30 Wm-2, corresponding to an evaporation rate of ∼ 1 mmw.e.d-1, even during the severe July 2022 dry spell. This value is in the range of evaporation rates reported for debris-covered glaciers of 0.6–2.8 mmw.e.d-1 . However, note that our evaporative flux estimate QLE might be an upper bound, in particular during the July 2022 dry spell. The moisture source for evaporative QLE is in the AL, not at the surface (except during precipitation events). However, our bulk parameterisation (Cbt; Eq. ) ignores the additional resistance to vapour transport imposed by the blocky layer between the specific humidity measurement in the cavity roof and the atmosphere (qsa is measured 0.7 m beneath the terrain surface; Fig. e). The neglected resistance to vapour transport in the coarse blocky AL during moisture-limited evaporation stages, when moisture is drawn from deeper levels and governed by vapour diffusion through the porous AL , might lead to an overestimation of QLE. The longer the dry spell lasts, the deeper in the AL the moisture is drawn from (Sect. ), and the more important this effect of vapour transport resistance becomes. This likely explains why the specific humidity in the cavity roof is almost always (down to sub-hourly timescales) higher than that in the atmosphere despite the relentless mixing by ventilation (moisture surplus relative to the atmosphere; Sect. ), and it possibly accounts for the negative July 2022 SEB imbalance (Fig. ). The upwards vapour transfer in the comparatively large and strongly ventilated cavity during the dry spell might be overly efficient compared with that in the surrounding AL. This might have led to a non-representatively high specific humidity qsa in the cavity roof and an overestimated evaporative flux QLE.

Our mean (median) values of the filtered aerodynamic momentum roughness lengths z0m of 19 to 7 cm (23–8 cm) agree with the values of and of 18 and 7 cm for the snow-free and snow-covered Murtèl surface, respectively (Fig. ). The scatter range of ∼ 2 orders of magnitude is similar to that in other studies . The absolute values are at the upper limit of what has been typically found on debris-covered glaciers or on Juvvasshøe by (5 cm), which is plausible given the rough terrain of the Murtèl rock glacier.

The calculated roughness length decreases slightly with increasing snow height by 0.1 cm per cm of snow (Fig. b). A similar but much stronger relation between snow height and roughness lengths was found on the Haut Glacier d'Arolla, where found z0m to decrease by 2 orders of magnitude with snow heights up to 3 m. This represents a much stronger relation than that on Murtèl, where z0m decreases by ∼0.5× over the observed range of snow thickness. With a maximum snow height of 120 cm (PERMA-XT measurement; Fig. ) or 200 cm (PERMOS) during the measurement period, the snow cover is thin compared with the terrain roughness (edge length of blocks ∼ 50 cm) and undulations. Both the comparatively high roughness length and the weak sensitivity to snow height might suggest that the aerodynamic roughness is largely controlled by the furrow-and-ridge micro-topography that is not smoothed out by the snow cover or by the few largest blocks that stick out of the snow cover rather than by the average block size. In forests, for comparison, the tallest trees of the canopy have a disproportionally large influence on the aerodynamic roughness .

For the bulk parameterisation, we linearly interpolate log⁡10{z0m} with snow height. Due to the lack of accurate humidity-corrected sonic temperature measurements, we cannot calculate z0h or z0q. We used the unknown ratio R^z0:=z0h/z0m=z0q/z0m as a calibration parameter and found R^z0=2×10-2. While the approximation z0h≈z0q seems applicable, z0m≈z0h is not compatible with our parameterisation (Table ). This agrees with previous studies on debris-covered glaciers and hummocky ice surfaces .