Abstract
That snow is warm is called a curiosity. This paper shows that what insulates is not the snow but the air inside it—so the same snow, once packed, conducts $\mathbf{6.3645}$ times as well. No new theorem or law is claimed.
First, fresh snow conducts at $\mathbf{0.069330}$ W/(m K), $\mathbf{1/32.02}$ of ice (Section 2).
Second, at density $100$, $\mathbf{0.8909}$ of the volume is air (Section 3).
Third, this is the core of the paper. $30$ cm of fresh snow equals $\mathbf{17.31}$ cm of glass wool, within the practical range of housing insulation (Section 2).
Fourth, packed to density $500$, the conductivity rises by a factor of $\mathbf{6.3645}$ (Section 2).
Fifth, the measured value lies between the series bound $\mathbf{0.029141}$ and the parallel bound $\mathbf{0.265258}$ (Section 3).
Sixth, the separator is whether the material insulates or the structure does (Section 4).
Keywords: snow, thermal conductivity, thermal insulation, air, rule of mixtures.
Scope (what is not claimed)
No new theorem or law is claimed—the density dependence of the thermal conductivity of snow and the series and parallel bounds of mixture rules are all standard. No building is designed—choice of insulation, installation and condensation control are not the subject. No snow shelter is described—safety, ventilation and avalanche hazard are not the subject. Vapour transport is excluded—sublimation and recondensation can carry heat through snow and are not always negligible, but only conduction is treated here. One empirical fit is fixed—a quadratic in density is used; coefficients vary between sources. Relation to earlier papers: Paper 450 showed that the same span of days carries maturity differing threefold—there too, one name covered two quantities. Paper 300 showed that common root and distinct root can be decided—“snow as a material” and “snow as a structure” are distinct roots by that criterion. Paper 447 showed that added workers flow out of the exit—there too, structure and not quantity was at work. What is added is giving the conductivity at four densities, giving the glass-wool equivalent as $17.31$ cm, giving the porosity as $0.8909$, bracketing the measurement between the series and parallel bounds, and placing the separator at material versus structure.
1. Notation and the Question
Setting: Take the thermal conductivity of snow as a quadratic in the density $\rho$ [kg/m$^3$],
$$k(\rho)=0.138-1.01\times10^{-3}\rho+3.233\times10^{-6}\rho^{2}$$with $k_{i}=2.22$ for ice and $k_{a}=0.026$ W/(m K) for air. A layer of thickness $d$ has thermal resistance $R=d/k$.
Observation 1.1 That snow is warm is called a curiosity. Snow is made of ice at $0 ^\circ$C—and the intuition says a cold thing cannot warm.
Observation 1.2 But it does not warm; it fails to let heat out. The question is not temperature but conductivity—and that is set by structure, not by what snow is made of.
Question: How much does it insulate, and what decides that.
2. Fresh Snow Conducts a Thirty-Second of Ice
Computation 2.1: Take the conductivity at four densities.
| Density [kg/m$^3$] | Conductivity [W/(m K)] |
|---|---|
| $100$ (fresh) | $\mathbf{0.069330}$ |
| $200$ | $0.065320$ |
| $300$ | $0.125970$ |
| $500$ (packed) | $\mathbf{0.441250}$ |
Observation 2.2 Fresh snow conducts $\mathbf{1/32.02}$ of ice. $0.069330$ against $2.22$—a factor of $32$ between two things made of the same substance.
Observation 2.3 Packing makes it conduct. $0.441250$ at density $500$, $\mathbf{6.3645}$ times fresh snow—tread the snow down and the insulation falls.
Observation 2.4 And still it is under half of ice. $0.441250$ against $1.11$—until it is fully compacted, air remains.
Observation 2.5 But between $100$ and $200$ in the table the value dips. From $0.069330$ to $0.065320$—this quadratic has a minimum of $\mathbf{0.059118}$ at density $\mathbf{156.2017}$.
Observation 2.6 That is an artefact of the fit, not a property of snow. Fitting a quadratic reverses the direction at the edge of its range—“packing makes it conduct” holds only above $156$; below that the fit guarantees nothing.
Observation 2.7 This is where the form of the fit can reverse the conclusion. Read as monotone, it says the opposite across $100$ to $200$—it is worth checking numerically that one has not stepped outside the fit.
Computation 2.8: Convert the resistance of $30$ cm of fresh snow into a thickness of housing insulation.
$4.3271$ m$^2$K/W, which as glass wool ($0.040$ W/(m K)) is $\mathbf{0.1731}$ m $=\mathbf{17.31}$ cm.
Observation 2.9 Almost exactly the thickness put into a house wall. Between $10$ and $20$ cm—$30$ cm of snow is insulation in fact, not in metaphor.
Observation 2.10 Against concrete the ratio is $\mathbf{23.08}$. At the same $30$ cm—here I first estimated “over $40$ times” and was wrong; the computed value is $23.08$.
3. Porosity 0.8909 — What Insulates Is the Air
Computation 3.1: Take the porosity of snow at density $100$.
$1-100/917=\mathbf{0.8909}$: $89$ per cent of the volume is air.
Observation 3.2 It is mostly air. Ice is only $11$ per cent—what is called “snow” is, in substance, air that holds its shape.
Computation 3.3: Bracket it between the two mixture bounds.
| Model | Conductivity [W/(m K)] |
|---|---|
| Series (lower) | $0.029141$ |
| Measured (fit) | $\mathbf{0.069330}$ |
| Parallel (upper) | $0.265258$ |
Observation 3.4 The measurement lies between them. $2.3792$ times the lower, $0.2614$ of the upper—within what two components, ice and air, can account for.
Observation 3.5 Sitting near the lower bound means the heat is going around the ice skeleton. In parallel the ice would be a short cut, and it is not—disconnected grains of ice float in the air.
Observation 3.6 Packing connects the ice and moves it towards the upper bound. This is why the quadratic climbs steeply with density—a factor of $6.3645$ from $100$ to $500$ is not a linear rise.
4. The Separator
The separator: does the material insulate, or the structure.
Observation 4.1 If the material insulates, the same substance always insulates the same. Ice is ice—one number, $2.22$, suffices.
Observation 4.2 If the structure insulates, the same substance varies with its arrangement. A factor of $32$, from $0.069330$ to $2.22$—the name of the material does not settle how heat passes.
Observation 4.3 These are distinct roots by the criterion of Paper 300. Both are called “the conductivity of snow”—one a property of a substance, one of a substance paired with a density.
Observation 4.4 The same shape as Paper 450. There, “$28$ days” was not a quantity without a temperature—here “snow” is not a quantity without a density. In both, the name does not name the quantity.
Observation 4.5 So designing insulation is not choosing a material but trapping air. Glass wool, foam and down work on the same principle—a structure that holds air still.
Observation 4.6 To be honest—vapour transport is not included. Sublimation and recondensation carry heat through snow and are not negligible under a steep gradient—only conduction has been looked at.
Observation 4.7 One last thing—the coefficients of the fit vary between sources. The direction—that it moves strongly with density—is firmer than the factor of $32$ itself—and the conclusion that structure does the insulating does not depend on the choice of coefficients.
5. Roll Call and Closing
- Fresh snow is $0.069330$ W/(m K) (Computation 2.1): $1/32.02$ of ice.
- $0.441250$ at density $500$ (Computation 2.1): $6.3645$ times as conductive.
- Still under half of ice (Observation 2.4): air remains.
- $30$ cm equals $17.31$ cm of glass wool (Computation 2.8): the core.
- $23.08$ times concrete (Observation 2.10): my estimate was wrong.
- Porosity $0.8909$ (Computation 3.1): mostly air.
- Series $0.029141$, parallel $0.265258$ (Computation 3.3): measurement between.
- Near the lower bound (Observation 3.5): the ice is not connected.
- The separator is material or structure (Observation 4.3): distinct roots by Paper 300.
- Insulation is trapped air (Observation 4.5): not a choice of material.
Closing: That snow is warm is called a curiosity—snow is made of ice at $0 ^\circ$C, and the intuition says a cold thing cannot warm. But it does not warm; it fails to let heat out. The question is not temperature but conductivity, and that is set by structure, not by what snow is made of. The amount of insulation is large—fresh snow conducts at $\mathbf{0.069330}$ W/(m K), which is $\mathbf{1/32.02}$ of ice: a factor of $32$ between two things made of the same substance. $30$ cm has a resistance of $4.3271$ m$^2$K/W, which as glass wool is $\mathbf{17.31}$ cm—almost exactly the thickness put into a house wall, so $30$ cm of snow is insulation in fact, not in metaphor. Against concrete the ratio is $\mathbf{23.08}$; here I first estimated “over $40$ times” and was wrong. And packing makes it conduct: $0.441250$ at density $500$, $\mathbf{6.3645}$ times fresh snow, so treading the snow down costs the insulation. What insulates is the air—at density $100$ the porosity is $\mathbf{0.8909}$, so $89$ per cent of the volume is air and only $11$ per cent is ice. What is called “snow” is, in substance, air that holds its shape. Bracketed by the mixture bounds—series $0.029141$, measured $0.069330$, parallel $0.265258$—it lies between them, within what ice and air alone can account for. Sitting near the lower bound means the heat goes around the ice skeleton: in parallel the ice would be a short cut, and it is not, because disconnected grains of ice float in the air. Packing connects them and moves the value towards the upper bound, which is why a factor of $6.3645$ from $100$ to $500$ is not a linear rise. One thing separates them—does the material insulate, or the structure. If the material insulates, the same substance always insulates the same and one number, $2.22$, suffices. If the structure insulates, the same substance varies with its arrangement across a factor of $32$, and the name of the material does not settle how heat passes. They are distinct roots by the criterion of Paper 300, one a property of a substance and one of a substance paired with a density. It is the same shape as Paper 450, where “$28$ days” was not a quantity without a temperature: here “snow” is not a quantity without a density. So designing insulation is not choosing a material but trapping air—glass wool, foam and down all work by holding air still. To be honest—vapour transport is not included. Sublimation and recondensation carry heat through snow and are not negligible under a steep gradient; only conduction has been looked at. One last thing—the coefficients of the fit vary between sources. The direction is firmer than the factor of $32$ itself, and the conclusion that structure does the insulating does not depend on the choice of coefficients.
References
- Sturm, M., Holmgren, J., König, M., & Morris, K. (1997). The Thermal Conductivity of Seasonal Snow. Journal of Glaciology.
- Yen, Y.-C. (1981). Review of Thermal Properties of Snow, Ice and Sea Ice. CRREL Report 81-10.
- Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism. Clarendon Press.
- Calonne, N., Flin, F., Morin, S., Lesaffre, B., du Roscoat, S. R., & Geindreau, C. (2011). Numerical and Experimental Investigations of the Effective Thermal Conductivity of Snow. Geophysical Research Letters, 38, L23501.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Add Too Many Workers and the Workers Start Leaving [Paper 447]. Zenodo. 10.5281/zenodo.22732494
- Yamagishi, Y. (2026). Twenty-Eight Days in Winter Is Not Twenty-Eight Days in Summer [Paper 450]. Zenodo. 10.5281/zenodo.22732716
Reference
Yamagishi, Y. (2026). What Insulates Is Not the Snow but the Air It Holds [Paper 451]. YOI: https://www.yamagishiyuuki.com/yoi/451
DOI: 10.5281/zenodo.22732772
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 451. This paper is a preprint and has not been peer reviewed.