Non-Oberbeck-Boussinesq effects in coldwater

Gustavo Estay , Daisuke Noto , Hugo N. Ulloa

Authors on Pith no claims yet

Pith reviewed 2026-05-07 17:53 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cond-mat.mtrl-sciphysics.geo-ph

keywords convectioneffectsnumberclassicalmaterialpropertiesheathere

0 comments

The pith

Non-Oberbeck-Boussinesq effects in near-freezing water lower mean temperature, break mean-profile symmetry, shift critical Rayleigh number slightly, and preserve classical Nu and Re scalings after correction at intermediate Prandtl number.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Water near 0 °C has an unusual density curve that peaks at 4 °C, plus rising viscosity and falling thermal conductivity as it cools. Standard convection models ignore these changes and assume constant properties plus a straight-line density-temperature relation. The authors instead keep the full temperature dependence inside the equations and simulate convection between two plates, one cold and one warmer, using direct numerical simulation. They observe that the fluid as a whole sits at a lower average temperature than in the constant-property case and that the temperature profile is no longer symmetric about the mid-plane. The onset of convection also moves to a slightly higher Rayleigh number. Once this new onset value is used, the heat transport (Nusselt number) and the flow speed (Reynolds number) still follow the same power-law relations that have been known for ordinary fluids. The Reynolds-number scaling that normally appears only at very high viscosity ratios here appears at a moderate viscosity ratio around 10. These changes matter for any body of water that is close to freezing and bounded by ice.

Core claim

We show that non-Oberbeck--Boussinesq effects lower the mean fluid temperature relative to the standard case and break the classical symmetry of the mean temperature profile. [...] After accounting for this shift, the nondimensional heat transfer rate, Nu, follows the classical scaling with supercriticality, while Re remains consistent with the Grossmann--Lohse unifying theory, Re∝(Ra−Ra_c)^{1/2} for low-Ra convection (regime I_u) and Re∝(Ra−Ra_c)^{4/7} at high-Ra (regime III_u).

Load-bearing premise

The specific functional forms chosen for ρ(T), μ(T), and k(T) near 0 °C, together with the assumption that the direct numerical simulations fully resolve all relevant scales and boundary layers without numerical artifacts or insufficient domain size, are sufficient to capture the reported symmetry breaking and the small Ra_c shift.

Figures

Figures reproduced from arXiv: 2604.24979 by Daisuke Noto, Gustavo Estay, Hugo N. Ulloa.

**Figure 1.** Figure 1: Temperature dependence of material properties for cold water, including (a) density, (b) dynamic viscosity and (c) thermal conductivity. Black dashed lines correspond to the models described in § 2.1, while key values are highlighted with diamonds, considering example values of 𝑇𝑏, 𝑇𝑟 and 𝑇𝑡 . Actual values of water properties are shown with continuous grey lines. The maximum relative error of each model i… view at source ↗

**Figure 2.** Figure 2: Instantaneous snapshots of the temperature field 𝜃 for cases with variable material properties for different values of the Rayleigh number: (𝑎) Ra = 2.15 × 103 , (𝑏) Ra = 2.15 × 105 , (𝑐) Ra = 4.64 × 107 . 4. Results We illustrate different convective regimes in figure 2 through instantaneous snapshots of the temperature field 𝜃 for cases with variable material properties. For Ra = 2.15×103 , the flow exhi… view at source ↗

**Figure 3.** Figure 3: (𝑎) Mean temperature 𝜃𝑚 as a function of the Rayleigh number Ra for simulations with CMP (red circles) and VMP (blue diamonds). Vertical bars denote temporal variability through standard deviation at steady state. The horizontal dashed lines indicate the mean temperature for the conductive state for cases with CMP (red) and VMP (blue). Cases that are also shown in figure 2 are highlighted with a grey circl… view at source ↗

**Figure 4.** Figure 4: (𝑎) Heat transfer rate represented by ‘Nu − 1’ as a function of ‘Ra − Ra𝑐’ for VMP (blue diamonds) and CMP (red circles) cases. Solid blue lines represent power-law fits for the VPM case across low and high Ra regions, whereas the dashed lines correspond to their extrapolation. In the fully steady convective regime, the fit obtained is Re ∝ (Ra − Ra𝑐); in the turbulent convective regime, the fitting gives … view at source ↗

**Figure 5.** Figure 5: Horizontally-averaged temperature, buoyancy, and temperature variance for increasing Ra, showing reduced upper-lower asymmetry and an upward shift of the zero-crossing of the mean temperature ⟨𝜃⟩𝑥 𝑦, from positive to negative bulk mean temperature. The intersection between ⟨𝑏⟩𝑥 𝑦 and ⟨𝜃˜2 ⟩𝑥 𝑦 that determines the change in temperature sign is highlighted with black circles. The neutrally buoyant height, 𝑧𝑛… view at source ↗

**Figure 6.** Figure 6: Panels from (𝑎) to (𝑑) show the individual and joint effects of variable conductivity and viscosity in the laterally-averaged temperature profile, with respect to the CMP case (5.1), for different Rayleigh numbers. Panel (𝑒) shows the relative effect of variable conductivity (𝑟𝑘 ) and viscosity (𝑟𝜇) in the temperature increase with respect to their joint effect (5.2), as a function of the Rayleigh number. … view at source ↗

read the original abstract

Water exhibits an anomalous nonlinear temperature-density ($\rho$-$T$) relation as it approaches freezing, along with an increase in viscosity, and a decrease in thermal conductivity. These departures from the standard Oberbeck--Boussinesq approximation, which assumes constant material properties and a linear $\rho$-$T$ relation, can modify convection in ice-bounded aquatic systems, yet their effects remain unexplored. Here, we examine these effects via the canonical Rayleigh--B\'enard convection framework using direct numerical simulations. We show that non-Oberbeck--Boussinesq effects lower the mean fluid temperature relative to the standard case and break the classical symmetry of the mean temperature profile. The magnitude of this symmetry breaking depends on both the Rayleigh number $Ra$ and the temperature-dependent material properties retained in the governing equations. We further identify a small but measurable shift in the critical Rayleigh number, $Ra_c$. After accounting for this shift, the nondimensional heat transfer rate, $Nu$, follows the classical scaling with supercriticality, while $Re$ remains consistent with the Grossmann--Lohse unifying theory, $Re\propto (Ra-Ra_c)^{1/2}$ for low-$Ra$ convection (regime $\mathrm{I}_u$) and $Re\propto (Ra-Ra_c)^{4/7}$ at high-$Ra$ (regime $\mathrm{III}_u$). Unlike the classical expectation that the latter scaling arises at high Prandtl number, here it is obtained at an intermediate Prandtl number, $Pr\sim 10$. Our results establish how near-freezing material anomalies affect both local and global properties of convection, with implications for heat distribution and mixing in cryospheric liquid waters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds that cold-water non-Oberbeck-Boussinesq effects break mean temperature symmetry and shift Ra_c slightly, after which Nu and Re follow standard scalings with the 4/7 Re exponent appearing at Pr~10.

read the letter

The core result is that variable density, viscosity, and conductivity near 0°C lower the mean temperature, break the usual symmetric mean profile, and produce a small Ra_c increase. After correcting for the shift, Nu tracks supercriticality as in the constant-property case, and Re matches Grossmann-Lohse regime I_u (exponent 1/2) at low Ra and regime III_u (exponent 4/7) at higher Ra, but at an intermediate Prandtl number around 10 instead of the high-Pr regime where that scaling normally appears. The symmetry breaking grows with Ra and with the strength of the property variations. This is new: prior work has not combined full temperature-dependent properties in this temperature window and reported these quantitative outcomes together. The study does a clean job of isolating the effect, running the DNS in the standard Rayleigh-Bénard geometry, and directly comparing the corrected global quantities to existing theory. That makes the claim that the scalings survive the non-Boussinesq corrections easy to check. The main soft spot is numerical resolution. With T-dependent viscosity and conductivity the boundary layers differ in thickness, so a uniform grid risks under-resolving the colder, more viscous side at high Ra. The abstract does not mention grid-convergence checks, boundary-layer point counts, or local Kolmogorov-scale comparisons, which leaves open the possibility that the apparent 4/7 scaling is influenced by under-resolution rather than being a genuine test of the theory. The chosen functional forms for the properties are reasonable but specific; modest changes in those fits could alter the size of the symmetry breaking. This work is aimed at researchers modeling convection in near-freezing lakes, under-ice waters, or other cryospheric settings where the 4°C density maximum matters. A reader who needs concrete corrections to heat-flux estimates or regime maps for variable-property convection will find usable numbers here. It is worth sending to peer review because the setup is straightforward, the comparison to theory is direct, and the potential implications are clear, provided the authors supply the missing resolution diagnostics and any error bars on the fitted exponents.

Referee Report

2 major / 2 minor

Summary. The manuscript examines non-Oberbeck-Boussinesq effects in Rayleigh-Bénard convection for water near freezing using direct numerical simulations. It finds that variable density, viscosity, and thermal conductivity lower the mean temperature, break the symmetry of the mean temperature profile, and cause a small shift in the critical Rayleigh number Ra_c. After correcting for the Ra_c shift, the Nusselt number follows the classical scaling with supercriticality, and the Reynolds number agrees with the Grossmann-Lohse theory in regimes I_u and III_u at Pr ≈ 10.

Significance. This study highlights the importance of material property variations in convection near 0°C, with potential implications for heat transport in cryospheric environments. The demonstration that GL theory holds after a small correction, even at intermediate Prandtl numbers, is a valuable extension. The use of DNS to capture these effects provides a solid foundation, though numerical validation is needed for full confidence.

major comments (2)

The manuscript does not specify the grid resolution, the number of grid points within the thermal and viscous boundary layers, or any grid-convergence tests. Given that μ(T) and k(T) vary with temperature, the boundary layers on the cold and warm sides have different thicknesses and properties; uniform grids may under-resolve the more viscous cold side, potentially affecting the accuracy of the Re measurements used to claim the 1/2 and 4/7 scalings.
The fitted exponents for Re ∝ (Ra - Ra_c)^β are presented without error bars or details on the fitting procedure and the range of Ra used. This makes it difficult to assess how well the data support the specific exponents from GL theory, especially since the Ra_c correction is small.

minor comments (2)

The regimes I_u and III_u are referenced without a brief definition or citation to the Grossmann-Lohse paper in the abstract, which could help readers unfamiliar with the unifying theory.
Some notation for the temperature-dependent properties could be clarified with explicit functional forms or references to the specific models used for ρ(T), μ(T), k(T) near 0°C.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the paper relies on the standard incompressible Navier-Stokes equations with temperature-dependent coefficients taken from literature, plus the usual Boussinesq buoyancy term modified by the full ρ(T) curve. No new entities are introduced.

axioms (2)

domain assumption The fluid remains incompressible except for the buoyancy term that uses the full nonlinear ρ(T) relation.
Standard for variable-property convection studies; invoked to justify the governing equations.
domain assumption Direct numerical simulation at the reported Rayleigh numbers fully resolves all boundary layers and turbulent structures.
Required for the claimed accuracy of mean profiles and Nu, Re values.

pith-pipeline@v0.9.0 · 5629 in / 1695 out tokens · 46535 ms · 2026-05-07T17:53:01.043964+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references

[1]

Ahlers, Guenter , Brown, Eric , Araujo, Francisco Fontenele , Funfschilling, Denis , Grossmann, Siegfried & Lohse, Detlef 2006 Non-Oberbeck--Boussinesq effects in strongly turbulent Rayleigh--B \'e nard convection . J. Fluid Mech. 569 , 409--445

2006
[2]

Allum, Donovan JM & Stastna, Marek 2024 Simulations of buoyant flows driven by variations in solar radiation beneath ice cover . Phys. Rev. Fluids 9 (6), 063501

2024
[3]

Allum, Donovan JM & Stastna, Marek 2025 Radiatively Driven Convection: A Comparison Between Two-and Three-Dimensional Simulations . Atmos. Ocean 63 (3), 159--171

2025
[4]

Bellincioni, Edoardo , Lohse, Detlef & Huisman, Sander G 2025 Melting of floating ice cylinders in fresh and saline environments . J. Fluid Mech. 1019 , A29

2025
[5]

Berhanu, Michael , Dawadi, Amit , Chaigne, Martin , Jovet, J \'e r \^o me & Kudrolli, Arshad 2026 Self-propulsion of floating ice blocks caused by melting in water . Phys. Rev. Fluids 11 (3), 033802

2026
[6]

Inland Waters 9 (2), 142--161

Bouffard, Damien , Zdorovennova, Galina , Bogdanov, Sergey , Efremova, Tatyana , Lavanchy, S \'e bastien , Palshin, Nikolay , Terzhevik, Arkady , Vinn , Love R man , Volkov, Sergey , W \"u est, Alfred & others 2019 Under-ice convection dynamics in a boreal lake . Inland Waters 9 (2), 142--161

2019
[7]

Boussinesq, Joseph 1903 Th \'e orie analytique de la chaleur: mise en harmonie avec la thermodynamique et avec la th \'e orie m \'e canique de la lumi \`e re\/ , , vol. 2 . Gauthier-Villars

1903
[8]

, Vasil, Geoffrey M

Burns, Keaton J. , Vasil, Geoffrey M. , Oishi, Jeffrey S. , Lecoanet, Daniel & Brown, Benjamin P. 2020 Dedalus: A flexible framework for numerical simulations with spectral methods . Phys. Rev. Res. 2 , 023068

2020
[9]

Courier Corporation

Chandrasekhar, Subrahmanyan 2013 Hydrodynamic and hydromagnetic stability\/ . Courier Corporation

2013
[10]

Couston, Louis-Alexandre 2021 Turbulent convection in subglacial lakes . J. Fluid Mech. 915 , A31

2021
[11]

Cross, MC 1980 Derivation of the amplitude equation at the Rayleigh-B \'e nard instability . Phys. Fluids 23 (9), 1727--1731

1980
[12]

Cross, Mark C & Hohenberg, Pierre C 1993 Pattern formation outside of equilibrium . Rev. Mod. Phys. 65 (3), 851

1993
[13]

Debenedetti, Pablo G 2003 Supercooled and glassy water . J. Phys. Condens. Matter 15 (45), R1669--R1726

2003
[14]

Energy Convers

Eggimann, Sven , Vivian, Jacopo , Chen, Ruihong , Orehounig, Kristina , Patt, Anthony & Fiorentini, Massimo 2023 The potential of lake-source district heating and cooling for european buildings . Energy Convers. Manag. 283 , 116914

2023
[15]

PNAS Nexus p

Estay, Gustavo , Noto, Daisuke & Ulloa, Hugo N 2026 Under-ice convective regimes driven by sunlight and sediment temperature control water--ice heat flux . PNAS Nexus p. pgag045

2026
[16]

Earth Surf

Fern \'a ndez, Roberto & Parker, Gary 2021 Laboratory observations on meltwater meandering rivulets on ice . Earth Surf. Dyn. 9 (2), 253--269

2021
[17]

Water Resour

Fink, Gabriel , Schmid, Martin & W \"u est, Alfred 2014 Large lakes as sources and sinks of anthropogenic heat: Capacities and limits . Water Resour. Res. 50 (9), 7285--7301

2014
[18]

Grace, Andrew P , Fogal, Alexander & Stastna, Marek 2023 Restratification in late winter lakes induced by cabbeling . Geophys. Res. Lett. 50 (14), e2023GL103402

2023
[19]

Grossmann, Siegfried & Lohse, Detlef 2000 Scaling in thermal convection: a unifying theory . J. Fluid Mech. 407 , 27--56

2000
[20]

Grossmann, Siegfried & Lohse, Detlef 2001 Thermal convection for large Prandtl numbers . Phys. Rev. Lett. 86 (15), 3316

2001
[21]

Holzbecher, Ekkehard 1997 Numerical studies on thermal convection in cold groundwater . Int. J. Heat Mass Transf. 40 (3), 605--612

1997
[22]

Horn, Susanne , Shishkina, Olga & Wagner, Claus 2013 On non-Oberbeck--Boussinesq effects in three-dimensional Rayleigh--B \'e nard convection in glycerol . J. Fluid Mech. 724 , 175--202

2013
[23]

Huang, Ruiqi , Ouyang, Zhen & Ding, Zijing 2025 Linear and nonlinear dynamics in radiatively forced cold water . J. Fluid Mech. 1011 , A24

2025
[24]

Food Res

Jiang, Ling , Liu, Donghong , Wang, Wenjun , Lv, Ruiling , Yu, Songfeng & Zhou, Jianwei 2025 Advancements and perspectives of novel freezing and thawing technologies effects on meat: A review . Food Res. Int. 204 , 115942

2025
[25]

Johnson, Bobae , Weady, Scott , Zhang, Zihan , Kim, Alison & Ristroph, Leif 2025 Shape evolution and capsize dynamics of melting ice . Phys. Rev. Fluids 10 (9), 093801

2025
[26]

Lam, Siu , Shang, Xiao-Dong , Zhou, Sheng-Qi & Xia, Ke-Qing 2002 Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh-B \'e nard convection . Phys. Rev. E 65 (6), 066306

2002
[27]

Leung, Michael , Ching, Wing-Han , Leung, Dennis YC & Lam, Gabriel CK 2007 Fluid dynamics and heat transfer in cold water thawing . J. Food Eng. 78 (4), 1221--1227

2007
[28]

Magnani, Marta , Musacchio, Stefano , Provenzale, Antonello & Boffetta, Guido 2024 Convection in the active layer speeds up permafrost thaw in coarse-grained soils . Phys. Rev. Fluids 9 (8), L081501

2024
[29]

Manga, Michael & Weeraratne, Dayanthie 1999 Experimental study of non-Boussinesq Rayleigh--B \'e nard convection at high Rayleigh and Prandtl numbers . Phys. Fluids 11 (10), 2969--2976

1999
[30]

2026 a\/ Effects of variable material properties in coldwater convection

Noto, Daisuke & Ulloa, Hugo N. 2026 a\/ Effects of variable material properties in coldwater convection . Under review in Phys. Rev. Fluids

2026
[31]

2026 b\/ Melting dynamics of freely floating ice in calm waters

Noto, Daisuke & Ulloa, Hugo N. 2026 b\/ Melting dynamics of freely floating ice in calm waters . Sci. Adv. 12 (5), eadw3368

2026
[32]

U ber die w \

Oberbeck, Anton 1879 \"U ber die w \"a rmeleitung der fl \"u ssigkeiten bei ber \"u cksichtigung der str \"o mungen infolge von temperaturdifferenzen . Annalen der Physik 243 (6), 271--292
[33]

Olsthoorn, Jason , Tedford, Edmund W & Lawrence, Gregory A 2021 The cooling box problem: convection with a quadratic equation of state . J. Fluid Mech. 918 , A6

2021
[34]

2021 Non-Boussinesq convection at low Prandtl numbers relevant to the Sun

Pandey, Ambrish , Schumacher, J\"org & Sreenivasan, Katepalli R. 2021 Non-Boussinesq convection at low Prandtl numbers relevant to the Sun . Phys. Rev. Fluids 6 , 100503

2021
[35]

Pramana 87 (1), 13

Pandey, Ambrish , Verma, Mahendra K , Chatterjee, Anando G & Dutta, Biplab 2016 Similarities between 2D and 3D convection for large Prandtl number . Pramana 87 (1), 13

2016
[36]

Pegler, Samuel S & Wykes, Megan S Davies 2021 The convective stefan problem: shaping under natural convection . J. Fluid Mech. 915 , A86

2021
[37]

Perga, ME , Syarki, M , Kalinkina, N & Bouffard, D 2020 A rotiferan version of the punishment of sisyphus? Ecology 101 (3), 1--4

2020
[38]

Toppaladoddi, Srikanth & Wettlaufer, John S 2018 Penetrative convection at high Rayleigh numbers . Phys. Rev. Fluids 3 (4), 043501

2018
[39]

Ulloa, Hugo N , Ram \'o n, Cintia L , Doda, Tomy , W \"u est, Alfred & Bouffard, Damien 2022 Development of overturning circulation in sloping waterbodies due to surface cooling . J. Fluid Mech. 930 , A18

2022
[40]

Ulloa, Hugo N , Winters, Kraig B , W \"u est, Alfred & Bouffard, Damien 2019 Differential heating drives downslope flows that accelerate mixed-layer warming in ice-covered waters . Geophys. Res. Lett. 46 (23), 13872--13882

2019
[41]

Van Der Poel, Erwin P , Stevens, Richard JAM & Lohse, Detlef 2013 Comparison between two-and three-dimensional Rayleigh--B \'e nard convection . J. Fluid Mech. 736 , 177--194

2013
[42]

Astrophys

Veronis, George 1963 Penetrative convection. Astrophys. J. 137 , 641

1963
[43]

Wan, Zhen-Hua , Wang, Qi , Wang, Ben , Xia, Shu-Ning , Zhou, Quan & Sun, De-Jun 2020 On non-Oberbeck–Boussinesq effects in Rayleigh–Bénard convection of air for large temperature differences . J. Fluid Mech. 889 , A10

2020
[44]

Fluid Mech

Weiss, Stephan , Emran, Mohammad S & Shishkina, Olga 2024 What Rayleigh numbers are achievable under Oberbeck--Boussinesq conditions? J. Fluid Mech. 986 , R2

2024
[45]

Wells, Mathew G & Wettlaufer, JS 2008 Circulation in Lake Vostok: A laboratory analogue study . Geophys. Res. Lett. 35 (3)

2008
[46]

Ocean Model

W \"u est, Alfred & Carmack, Eddy 2000 A priori estimates of mixing and circulation in the hard-to-reach water body of lake vostok . Ocean Model. 2 (1-2), 29--43

2000
[47]

Xia, Ke-Qing , Lam, Siu & Zhou, Sheng-Qi 2002 Heat-flux measurement in high-Prandtl-number turbulent Rayleigh-B \'e nard convection . Phys. Rev. Lett. 88 (6), 064501

2002
[48]

Yang, Bernard , Wells, Mathew G , McMeans, Bailey C , Dugan, Hilary A , Rusak, James A , Weyhenmeyer, Gesa A , Brentrup, Jennifer A , Hrycik, Allison R , Laas, Alo , Pilla, Rachel M & others 2021 A new thermal categorization of ice-covered lakes . Geophys. Res. Lett. 48 (3), e2020GL091374

2021
[49]

Yang, Bernard , Young, Joelle , Brown, Laura & Wells, Mathew 2017 High-frequency observations of temperature and dissolved oxygen reveal under-ice convection in a large lake . Geophys. Res. Lett. 44 (24), 12--218

2017
[50]

, Liu, Hao-Ran , Verzicco, Roberto & Lohse, Detlef 2023 Bistability in radiatively heated melt ponds

Yang, Rui , Howland, Christopher J. , Liu, Hao-Ran , Verzicco, Roberto & Lohse, Detlef 2023 Bistability in radiatively heated melt ponds . Phys. Rev. Lett. 131 , 234002

2023
[51]

PRX Energy 3 (4), 043006

Yang, Rui , Howland, Christopher J , Liu, Hao-Ran , Verzicco, Roberto & Lohse, Detlef 2024 Enhanced efficiency of latent heat energy storage by inclination . PRX Energy 3 (4), 043006

2024
[52]

Zhu, Xiaojue , Mathai, Varghese , Stevens, Richard JAM , Verzicco, Roberto & Lohse, Detlef 2018 Transition to the ultimate regime in two-dimensional Rayleigh-B \'e nard convection . Phys. Rev. Lett. 120 (14), 144502

2018
[53]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...
[54]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...