1) Seasonal frost accumulates behind boulders on Mars and shields the area from sunlight. As the sun rises higher in the sky, the temperature behind boulders rises rapidly, such as increasing by 100K within a sol. 2) While the melting point of water is reached, evaporative cooling prevents the formation of liquid water. Instead, briny solutions can form in favorable conditions with high energy input and where salts are present in the substrate. 3) For liquid water to exist, the atmospheric pressure would need to be much higher than the current levels on Mars to reduce evaporative cooling effects. Overall, the model shows that pure liquid water is not stable on the surface of Mars today, but br

1. Crocus Melting on Mars
Norbert Sch¨orghofer
Planetary Science Institute, Hawaii
Building on work by Gary Clow, Michael Hecht, Andrew Ingersoll
and others
N. Schorghofer. Mars: Quantitative evaluation of crocus melting
behind boulders. Astrophysical Journal 890, 49 (2020)

2. Planet Mars
• 5 mbar CO2 atmosphere, contains 1–2 µbar water vapor
• Rotation period 24.6 hours, Length of year 687 earth days
• Axis tilt (obliquity) 25◦
• Effective temperature 210 K (145–310 K)
The most earth-like planet in the solar system

3. Phase Diagrams for Mars
150 200 250 300
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
Temperature (K)
PartialPressure(Pa)
Mars average
Frostpoint
198 K
liquid
Total pressure
H2O
CO2
Triple point 273K
Melting Point (273 K) Frost Point (∼200 K)

4. Physical Barriers to Melting
Melting point (273 K) Frost point (200 K)
liquid water stable ice
In terms of vapor pressure:
pH2O(273K) ≈ 4000 × pH2O(200K)
611 Pa 0.1 − 0.2 Pa
⇒ rapid dispersal of vapor and loss of ice
• A source of H2O must be available (subsurface ice or seasonal
water frost)
Potential pathway to liquid water: Rapid change from frost point
to melting point (fast ⇒ small loss)
Atmospheric pressure 520 Pa (200–1200)
Triple point 611 Pa
• Enough energy must be available to compensate for evapo-
rative (sublimation) cooling (Ingersoll, 1970)

5. Role of Topography
summer frost in shadow (Mauna Kea, Hawaii; image: OMKM)

6. Low-Latitude Frost, Mars
2003−02−08 Ls=135°
40 km
N
*
East Longitude
Latitude
135 136 137 138 139
−37
−38
−39
−40
MOC HiRISE 43◦S NASA/JPL/UA
pole-facing slopes; temperature heterogeneity; CO2 frost
first day without CO2 frost = “crocus date”

7. Recurring Slope Lineae
3D topography greatly
affects temperature and
sublimation rate
Palikir Crater
41.6◦S, 202.3◦W
NASA/JPL/UA

8. Thermal Model for 3D Geometry
Qsolar
Qatm
Qfloor
σT4
conduction
T
Tfloor
3D Energy Balance on Slope:
• Direct solar irradiance (incidence angle)
• Subsurface conduction
• Terrain shadowing (horizons) – Multigrid acceleration
• Terrain irradiance (long-wavelength & short-wavelength)
These factors were considered by Hecht (2002), Kossacki &
Markiewicz (2004), and others, for gully alcoves.

9. Seasonal frost behind boulder

10. Model results for idealized boulder 30◦S

11. Temperature abruptly rises from 145K to 280K within one sol
0 45 90 135 180 225 270 315 360
0
100
200
300
Areocentric Longitude Ls ( ° )
CO
2
Mass(kg/m2)
WinterSolstice
Seasonal shadow
Crocusdate
flat ground
behind boulder
194 195 196 197 198
150
200
250
300
Areocentric Longitude Ls ( ° )
SurfaceTemperature(K)
flat ground
behind boulder
behind boulder w/ subl.
Frost point
Melting point
without evaporative cooling

12. Rate of free convection on Mars
Triple point of H2O: 611 Pa, 273.16 K
Total pressure of atmosphere: 520 Pa (200–1200 Pa)
Molar mass of CO2: 44
Molar mass of H2O: 18 (humid air is lighter, buoyant)
Near the melting point, humid air has a strong buoyancy effect
that leads to free convection and “evaporative” cooling.
The expression from Ingersoll (1970) is
Ec = 0.17ρwDm
g
ν2
∆ρ
ρ
1/3
Ec ... convective flux, Dm ... molecular diffusivity, g ... specific surface gravity,
ν ... kinematic viscosity, ρw, ∆ρ, ρ ... densities

13. Turbulent convection: Theory I
Molecular flux (index m) and Convective flux (index c)
Since the equations governing heat transport and mass transport
are mathematically equivalent, both transport coefficients are
given by the same function Φ:
κc = κmΦ(Gr, Pr) heat diffusivity
Dc = DmΦ(Gr, Sc) mass diffusivity
Gr ... Grashof number
Pr ... Prandtl number, Sc ... Schmidt number
In the turbulent regime at high Grashof number,
Φ(Gr, N) = C(Gr · N)1/3
Malkus exponent 1/3; universal prefactor C

14. Turbulent convection: Theory II
The Grashof number is defined by
Gr = L3 g
ν2
∆ρ
ρ
where g = 3.71 m/s2 is the specific surface gravity on Mars.
The formula for the mass flux becomes
Ec = CDmρw
g
ν2
∆ρ
ρ
1/3
ν
Dm
1/3
where Sc = ν/Dm.

15. Numerical value of the prefactor
Source C Sc1/3 C Pr1/3 C
mass transfer heat transfer
Mull & Reiher (1930)∗ (in air)
Jakob (1949) 0.068 in air
Ingersoll (1970) 0.17 on Mars
Fujii & Imura (1972)∗ (in water) 0.13
Holman (1990) quotes Fujii & Imura (1972) 0.13
Mills (2001) 0.14
Hecht (2002) 0.15 × 0.51/3 0.15
Incropera et al. (2007) x 0.15
∗original measurements.

16. Modification
The relative density difference between humid and dry air is
∆ρ
ρ
=
pw(1 − Mw/M1)
p0 − pw(1 − Mw/M1)
(old)
Sublimation rate must diverge when pw = p0. Modify,
∆ρ
ρ
=
pw(1 − Mw/M1)
p0 − pw
(new)
Near pw ≈ p0 similarity of heat and mass flux breaks down, so
modification is justified.

17. Evaporative Cooling by Free Convection
230 240 250 260 270
10
-6
10
-5
10
-4
10
-3
Temperature (K)
Sublimationrate(kg/m
2
s)
586 W/m
2
Ingersoll (1970)
New Parametrization
divergence at
pH2O = p0

18. 0 45 90 135 180 225 270 315 360
0
100
200
300
Areocentric Longitude Ls ( ° )
CO
2
Mass(kg/m2)
WinterSolstice
Seasonal shadow
Crocusdate
flat ground
behind boulder
194 195 196 197 198
150
200
250
300
Areocentric Longitude Ls ( ° )
SurfaceTemperature(K)
flat ground
behind boulder
behind boulder w/ subl.
Frost point
Melting point
With evaporative cooling 273 K is not reached within a few sol.
⇒ No liquid water on Mars, but brines form periodically

19. Favorable Energy Input
6 7 8 9 10 11 12
200
220
240
260
280
300
320
Local Time (hr)
Temperature(K)
without latent heat
p
0
=500 Pa, ζ=0
p
0
=500 Pa, ζ=2 mm
p
0
=1000 Pa, ζ=0
p
0
=1000 Pa, ζ=3 mm
6 7 8 9 10 11 12
0
0.2
0.4
0.6
0.8
1
Local Time (hr)
IceLoss(kg/m2)
p
0
=500 Pa, ζ=0
p
0
=500 Pa, ζ=2 mm
p
0
=1000 Pa, ζ=0
p
0
=1000 Pa, ζ=3 mm
Ice loss from morning until noon. Solar energy input corresponds
to the equator at perihelion and an albedo of 0.15.
Left: Equilibrium surface temperature as a function of local time.
Right: Ice loss as a function of local time; 1 kg/m2 ≈ 1 mm.
p0 ... atmospheric pressure
ζ ... thickness of overlying dust layer of micron-sized particles

20. Conclusions
Pathway from frost to brines (at mid-latitudes):
• Water frost accumulates seasonally behind boulders.
• Even seasonal CO2 frost accumulates behind boulders.
• After the CO2 frost disappears, temperature rises rapidly
(e.g., by 100 K within one sol).
• Evaporative cooling prevents melting of pure ice, but peak
temperatures of -10◦C are realistic ⇒ Brines form on salt-rich
substrate.