The electronics, manufactured by CAPACITEC, supplies sensors with a current of constant amplitude by controlling the AC voltage across a reference impedance fed with a stable “clock” oscillator of frequency $f \approx 15.6254$ kHz, thereby producing sensor and guard voltage amplitudes proportional to the impedance between sensor and ground (Louge et al., 1997). When the probe is exposed to air alone, its impedance $Z_0 = (2\pi f tC_0)^{-1}$ is only function of the capacitance $C_0$ between sensor and target. In humid sand, the impedance becomes $Z = [2\pi f C_0 (K''_e + tK'_e)]^{-1}$. By forming the ratio of guard voltage amplitude in air $V_0$ and in sand $V$, the probe then records the modulus $V_0/V = (K'^2_e + K''^2_e)^{1/2}$. Louge et al. (2021) offer algorithms to determine guard amplitude and phase with high accuracy. Using them, the phase yields $\tan\varphi \equiv K''_e/K'_e$ and, ultimately, as shown in Section 4, the acquired clock and guard signals provide sand bulk density and $\Omega$ simultaneously at each sensor (Louge, Valance, Babah, et al., 2010).

The PCB is backed by a rigid composite lance made of DELRIN™ plastic and a glass filler, with effective heat conductivity $\approx 0.35$ W/m°K. This precaution minimizes thermal interference with surrounding sands of similar conductivity $k_s \approx 0.49$ W/m K. (A metal probe, which we tried unsuccessfully, dried sand deeply by conducting heat from the surface at high noon.) The hard-plastic housing lance is tapered to an asymmetric thin edge with 10° angle to impart minimum sand compression in front of the sensors during probe insertion.

The top-rear of the lance supports a carefully guarded enclosure to shield 15 sensor wires connecting the PCB to a AMPHENOL multi-position circular connector plug. The latter serves as quick connection with a guarded coaxial cable carrying all 15 sensor wires to a guarded multiplexer. Upon receiving a five-bit binary address, the multiplexer operates relay switches to connect one of the sensors to the processing electronics, while guarding all other. Relays and digital electronics in the multiplexer are powered by a small battery. To avoid stray capacitances within the multiplexer, the DC voltage of this battery is made to “float” with respect to the ground reference of the probe, that is, these two voltages are strictly independent.

A compact Field Programmable Gate Array deployed in a NATIONAL INSTRUMENT cRIO controls the multiplexer by sending it a binary address through a NI-9401 module, consistent with a pre-programmed schedule. The cRIO acquires 150 clock and guard voltages simultaneously using two separate synchronous NI-9201 data acquisition cards at a rate of 350 kS/s. It also acquires data from the Kipp and Zonen radiometer and the temperature probe of Louge et al. (2013). Finally, the cRIO orchestrates data storage on a USB-A memory flash drive. After retrieval from the latter, a MATLAB algorithm described by Louge et al. (2021) calculates $K'_e$ and $K''_e$. The system is powered by a distant 12 V lead-acid battery recharged by a 1 kW solar panel, or by lithium-polymer battery packs buried nearby for minimal footprint.

## Appendix B: Size Distribution

In the film adsorption regime, spherical sand particles of diameter $d$ and material density $\rho_p$ at equilibrium with the surrounding RH$_e$ have masses $(\pi/6)d^3\rho_p$ of solid and $\pi d^2\ell_w\rho_w$ of liquid, thereby holding a solid-bound water mass fraction

$$ \Omega_{d,e} = \frac{\Omega_{1_d}}{[T^*\ln(1/RH_e)]^{1/3}} \approx \frac{\Omega_{1_d}}{\{T^*\ln[p_{\text{sat}}/(pM^*Y_e)]\}^{1/3}}, \quad (B1) $$

where

$$ \Omega_{1_d} = \frac{6\ell_0 \rho_w}{d \rho_p} \quad (B2) $$

is inversely proportional to grain diameter (Shahraeeni & Or, 2010). In these expressions, the subscripts $e$ and $d$ indicate, respectively, equilibrium and the particle size class of diameter $d$. To average $\Omega_{1_d}$ over the entire population, we use the normalized particle size distribution $f_M$, such that the elementary mass fraction of grains with diameter in the range $[d, d + dd]$ is $f_M dd$. The result

$$ \Omega_1 = \int_{d=0}^{\infty} \Omega_{1_d} f_M dd = 6\ell_0 \frac{\rho_w}{\rho_p} \int_{d=0}^{\infty} f_M \frac{dd}{d} = \frac{6\ell_0 \rho_w}{d_{2,3} \rho_p}, \quad (B3) $$
tially with depth. Toward explaining this, we write Eq. (21) in a dimensionless form denoted by asterisks,

$$ \frac{p^*}{T^*} \frac{\partial Y}{\partial t^*} + \frac{p^*}{T^*} u^* \frac{\partial Y}{\partial x^*} - \mathbb{L} \frac{\partial}{\partial x^*} \left[ T^{*n_d} \left( \frac{\partial Y}{\partial x^*} + a_T Y \frac{\partial \ln T^*}{\partial x^*} \right) \right] = - \frac{p^*}{T^*} \frac{(Y - Y_e)}{\tau^*} (1 - Y), \quad (46) $$

where $\tau^* \equiv \tau/J$. Here, the diffusion term involves a Soret effect with coefficient $a_T$ coupling variations of $T$ and $Y$, which Appendix E dismisses as unimportant.

In this equation, $Y_e$ is known in terms of the instantaneous $\Omega$ held by grains from Eqs. (3) and (7). On short time scales, it is therefore invariant as equilibrium cannot be restored fast enough. In this analysis, we consider waves of $Y$ traveling about its mean value $\bar{Y}$ at the dimensionless frequency $f^* \equiv fJ$,

$$ Y = \bar{Y} + \delta Y \exp[\iota (2\pi f^* t^* - k^* x^*)], \quad (47) $$

where $\iota^2 = -1$. Substituting this expression in Eq (46), expanding to leading order in $\delta Y/\bar{Y} \ll 1$, simplifying $(1-Y) \approx 1$, subtracting the time-averaged equation, applying derivatives of Eq. (47), and dividing by the exponential term and $\delta Y$, Eq. (46) yields the characteristic equation

$$ A_c k^{*2} - \iota B_c k^* + C_c = 0, \quad (48) $$

where

$$ A_c \equiv \mathbb{L} T^{*n_d} \quad (49) $$

$$ B_c \equiv \frac{p^*}{T^*} u^* - \mathbb{L} (n_d + a_T) T^{*n_d} \frac{\partial \ln T^*}{\partial x^*} $$

$$ C_c \equiv -a_T \mathbb{L} T^{*n_d} \left[ (n_d - 1) \left( \frac{\partial \ln T^*}{\partial x^*} \right)^2 + \frac{\partial \ln T^*}{\partial t^*} \right] + \frac{p^*}{T^*} [\iota 2\pi f^* + 1/\tau^*], $$

and $T^*$ and $u^*$ are sampled in the midst of the aeolian transport period at $t^* \approx 1.57$.

There are two solutions $k_r^{*\pm} = k_i^{*\pm} - \iota k_i^{*\pm}$ to quadratic Eq. (48), with real and imaginary parts $k_r^{*\pm}$ and $k_i^{*\pm}$,

$$ k^{*\pm} = \iota \frac{B_c}{2A_c} \pm \frac{1}{2A_c} \sqrt{-B_c^2 - 4A_c C_c}. \quad (50) $$

The two resulting traveling waves in $Y$ have the form $\delta Y \exp(k_i^{*\pm}x^*) \times \exp[\iota(2\pi f^*t^* - k_r^{*\pm}x^*)]$. Of these, only the $k_r^{*\pm}$ wave has $k_i^{*\pm} < 0$, and therefore vanishes as $x^* \to \infty$.

In the regime of rapid oscillations and slow return to equilibrium (Fig. 9), we match its decreasing amplitude along $x^*$

$$ \Delta Y = \delta Y \exp(-|k_i^{*+}|x^*) \quad (51) $$

to the exponential decay with characteristic length $\ell^* \equiv -1/k_i^{*+} = 0.28 \pm 0.05$, or $\ell = 4.9 \pm 0.9$ cm. This relatively short $\ell$ justifies why it is crucial to measure water mass fraction with a probe of narrow vertical spatial resolution.

Meanwhile, Eqs. (49)-(50) yield $k_i^{*+}$ in terms of the known parameters $\mathbb{L}$, $n_d$, and $a_T$, the measured variables $T^*$, $u^*$ and $p^*$, as well as $\tau^*$ and $f^*$. Therefore, they can be recast as the relation between $\tau^*$ and $f^*$ that yields the measured $\ell^*$. For our conditions, this relation is conveniently fitted as $\tau^* \approx \tau_1^* + \delta\tau^* / [(1/f^* - (1/f_1^*))]$, where $(1/f_1^*) \approx 0.0084$ (12 min), $\delta\tau^* \approx 4.2 \times 10^{-5}$ and $\tau_1^* \approx 0.01$ (14 min). Although we did not sample $Y$ fast enough to establish its high frequency content, an inspection of oscillations in the bistable $Y$-record (thin lines in Fig. 9) suggests that $0.013 \lesssim 1/f^* \lesssim 0.033$ (19 to 48 min), thus yielding $\tau^* = 0.015 \pm 0.004$ (22 $\pm$ 5 min).

Such time is much larger than the diffusion time $\tau_{\text{DIFF}} \sim 240$ µs estimated in Appendix G. Therefore, if the diffusion limit prevailed, sand grains would have ample time
which appears in the isotherm of Equation 3, involves the “Sauter” diameter $d_{2,3}$. It is obtained by relating $f_M = d^3 f_N / \int_0^\infty f_N d^3\mathbf{d}$ to the normalized population size distribution $f_N$. The notation $d_{2,3}$ adopts the convention of Babinsky and Sojka (2002),

$$d_{p,q} = \left[ \frac{\int_0^\infty d^p f_N \, dd}{\int_0^\infty d^q f_N \, dd} \right]^{1/(p-q)} \quad (B4)$$

Louge et al. (2013) provided moments of the measured distribution $f_M$, summarized in their Supporting Information.

## Appendix C: Isotherm Derivatives

We define

$$\begin{aligned} \frac{\partial \Omega_e}{\partial t} &\equiv \Omega_1 \left[ \frac{f_T}{T_{st}} \frac{\partial T}{\partial t} + f_Y \frac{\partial Y}{\partial t} + \frac{f_p}{p_{st}} \frac{\partial p}{\partial t} \right] \\ &= \left[ \left( \frac{\partial \Omega_e}{\partial T} \right)_{RH} + \left( \frac{\partial \Omega_e}{\partial RH} \right)_T \times \left( \frac{\partial RH}{\partial T} \right)_{Y,p} \right] \frac{\partial T}{\partial t} \\ &+ \left[ \left( \frac{\partial \Omega_e}{\partial RH} \right)_T \times \left( \frac{\partial RH}{\partial Y} \right)_{p,T} \right] \frac{\partial Y}{\partial t} \\ &+ \left[ \left( \frac{\partial \Omega_e}{\partial RH} \right)_T \times \left( \frac{\partial RH}{\partial p} \right)_{T,Y} \right] \frac{\partial p}{\partial t}, \end{aligned} \quad (C1)$$

where subscripts indicate which variables are held constant in calculating the derivative. Differentiating Equation 3 and Equations 6 and 7, we find

$$\begin{aligned} \left(\frac{\partial RH}{\partial Y}\right)_{p,T} &= \left(\frac{p}{p_{\text{sat}}}\right) \frac{M^*}{\left[1 + Y(M^* - 1)\right]^2}, \\ \left(\frac{\partial RH}{\partial T}\right)_{Y,p} &= -\text{RH} \frac{T_{A_u}}{(T - T_{A_0})^2}, \\ \left(\frac{\partial RH}{\partial p}\right)_{T,Y} &= \frac{M^*}{p_{\text{sat}}} \times \frac{Y}{1 + Y(M^* - 1)}, \end{aligned} \quad (C2)$$

and

$$\begin{aligned} \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial T} \right)_{RH} &= - \frac{1}{3 T_{st} \ln^{1/3} (1/RH)} \left( \frac{T_{st}}{T} \right)^{4/3}, \\ \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial RH} \right)_T &= \frac{1}{3RH \ln^{4/3} (1/RH)} \left( \frac{T_{st}}{T} \right)^{1/3}, \end{aligned} \quad (C3)$$

which can be substituted into Equation C1 to find expressions for the dimensionless functions $f_T$, $f_Y$, and $f_p$ in terms of $Y$, $T$, and $p$. In particular,

$$f_Y = \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial Y} \right)_{T,p} = \frac{1/3}{Y [1 + Y(M^* - 1)] T^{*1/3} \ln^{4/3} (1/RH)} \approx \frac{1/3}{Y T^{*1/3} \ln^{4/3} [p_{\text{sat}} / (p M^* Y)]} \quad (C4)$$

Because grains of diameter $d$ have isotherms captured by Equation B1, they are individually subject to Equations C3 and C4, in which $\Omega_{1d}$ is substituted for its average $\Omega_1$ over the particle size distribution.

## Appendix D: Monin-Obukhov Similarity

In the spirit of Prandtl's mixing length hypothesis, Monin and Obukhov postulated that the turbulent transport of momentum is coupled to the fluxes of heat $\dot{q}''_{\text{wind}}$ and vapor mass $\dot{\Psi}''$ in the ABL through the similarity (Monin, 1970)
to return everywhere to an equilibrium with the surrounding vapor mass fraction. In that case, as section 7 showed, the right side of Eq. (46) would become $-\mathbb{R}\Omega_1 f_Y \partial Y / \partial t^*$, where temperature changes negligibly, $f_Y \propto 1/Y$, and $\mathbb{R} \equiv \rho_p \nu / [\rho_{st}(1 - \nu)]$. Then, the procedure leading to characteristic Eq. (48) would leave $A_c$ and $B_c$ unchanged, while replacing $C_c$ in Eq. (49) with

$$C'_c = \frac{p^*}{T^*} \iota 2\pi f^* + \mathbb{R}\Omega_1 \bar{f}_Y \left( \iota 2\pi f^* - \frac{\partial \ln \bar{Y}}{\partial t^*} \right), \quad (52)$$

an expression that no longer involves $\tau$. Using recorded $T^*$, $u^*$, $\bar{f}_Y$, and $\partial \ln \bar{Y} / \partial t^*$ at $t^* \simeq 1.6$, we find that $\ell^*$ would never exceed 0.045, which is much smaller than the observed $\ell^*$. Therefore, we conclude that evaporation is a kinetic-limited process in our case.

A common alternative approach to the kinetics of evaporation assumes that water in the adsorbed film is at thermodynamic equilibrium with the surrounding RH, while time-history is reintroduced through mass conservation and diffusion (Eq. 21). Such equilibrium approach to the evaporation rate, which is suitable for the drying of porous media such as wood or biomass, see for example Nasrallah and Perré (1988); Zhang and Datta (2004); Borujerdi et al. (2019), would fail for these hyper-arid sands.

Lastly, our determination of $\tau^* \simeq \tau_{HKS}^* = 0.015 \pm 0.004$ yields the accommodation coefficient of evaporation. Substituting the mean Sauter diameter $d_{2,3}$ for $d$ in Eq. (G1), and taking $\kappa_c$ to be on the same order, we find $\kappa_e = (2.3 \pm 0.5) \times 10^{-10}$. Such value is very small by the standards of pure, smooth surfaces. However, microscopic impurities, which are certain to contaminate desert sands, and interface coverage (Rubel & Gentry, 1984) are known to thwart interfacial mass transfer and reduce $\kappa_e$ by several orders of magnitude (Marek & Straub, 2001).

Coincidentally, this low probability is also consistent with an activation energy $\hat{E} = -\hat{R}T \ln \kappa_e = 54.6 \pm 0.6 \text{ kJ/mole}$, another common interpretation of the evaporation process (Rubel & Gentry, 1984; Y. Q. Li et al., 2001; Smith et al., 2006; Cappa et al., 2007; Tsukahara et al., 2009). This value is typical of the energy needed to release solid-bound water in solid fuels (Borujerdi et al., 2019) or powders (Prado & Vyazovkin, 2011) and, as expected, it is above, – but on the order of –, the latent heat $\simeq 43.9 \text{ kJ/mole}$ to vaporize unbound water.

In short, the characteristic decay depth of the evanescent waves suggests that evaporation from these sand grains is a kinetic-limited process. Although the latter may be interpreted as having a very low accommodation coefficient, it is more intuitive to regard it as an activated reaction.

## 13 Atmospheric boundary layer

The instantaneous evaporation flux exchanged with the atmospheric boundary layer (ABL) from the vadose zone is a crucial boundary condition for understanding the hydrology (Assouline & Kamai, 2019; Shao et al., 2021) and microbiology (Kidron & Starinsky, 2019) of hyper-arid regions. Because it is important to distinguish evaporation from transpiration when vegetation is involved (R. G. Anderson et al., 2017), flux techniques deployed in the ABL, such as eddy-covariance, are supplemented by carbon dioxide measurements (Scanlon & Kustas, 2010), sometimes involving stable isotopes (Griffis, 2013).

Measurements of the evaporation flux from within the sand surface are equally challenging. For example, in a fog desert, Feigenwinter et al. (2020) used an automated microlysimeter (ML) that recorded the mass of non-rainfall water (NRW) deposited in a sampling cup mounted flush with the surface. However, Kidron and Kronenfeld (2020) noted how the ML technique can overestimate the NRW amount, especially at night.
$$
\begin{align*}
\rho v_{\tau}^{2} &= \frac{\kappa z \rho v_{\tau}}{\phi_{m}} \frac{\partial U}{\partial z}, \\
\dot{q}_{\text{wind}}'' &= \frac{\kappa z \rho c_{p} v_{\tau}}{\phi_{h} \text{Pr}_{t}} \frac{\partial T}{\partial z}, \tag{D1} \\
\dot{\Psi}'' &= \frac{\kappa z \rho v_{\tau}}{\phi_{Y} \text{Sc}_{t}} \frac{\partial Y_{a}}{\partial z},
\end{align*}
$$

where $\kappa z$ is the mixing length, $c_p$ is specific heat at constant pressure, and $\text{Pr}_t$ and $\text{Sc}_t$ are, respectively, turbulent Prandtl and Schmidt numbers that Monin (1970) called “ratios of exchange coefficients” between heat and momentum or mass and momentum. Note that, for consistency with the internal flux, $\dot{\Psi}''$ and $\dot{q}''_{\text{wind}}$ in the ABL are positive downward. The quantities $\phi_m$, $\phi_h$, and $\phi_Y$ are heuristic functions of the relative altitude $z/L$ mentioned in Section 13. For heat transfer, Louge et al. (2013) adopted

$$
\phi_h = \begin{cases} 1 + a_{hN} z/L & , L > 0 \\ 1/(1 - a_{hD} z/L)^{1/2} & , L < 0 \end{cases} \tag{D2}
$$

with coefficients $a_{hN} \approx 7.8$ and $a_{hD} \approx 12$.

Making $\rho$, $T$, $v_\tau$, $\dot{q}''_{\text{wind}}$, and $z$ dimensionless with $\rho_{\text{st}}$, $T_{\text{st}}$, $(\alpha J)^{1/2}$, $\rho_{\text{st}} c_p T_{\text{st}} (\alpha J)^{1/2}$, and $(\alpha J)^{1/2}$, respectively, the unsteady heat balance in the ABL is

$$
\left( \frac{\text{Pr}_t e^z}{\kappa v_\tau^*} \right) \frac{\partial (\rho^* T^*)}{\partial t^*} = \frac{\partial}{\partial z} \left( \frac{\rho^*}{\phi_h} \frac{\partial T^*}{\partial z} \right), \quad (\text{D3})
$$

where we define $\xi \equiv \ln z$. Because $\rho^*T^* = p^*$ is invariant at constant atmospheric pressure $p^* \equiv p/p_{\text{st}}$, Equation D3 has no unsteady term, and the flux is independent of $z$. Integrating it between the surface at $z = z_0$ (index s) and the altitude $z_T$ where ambient temperature $T_T$ is measured, we find

$$
T^*(z^*) = T_s^* \exp \left[ \frac{\Phi_{2h} (z^*/L^*)}{\Phi_{2h} (z_T^*/L^*)} \ln \left( \frac{T_T^*}{T_s^*} \right) \right], \quad (\text{D4})
$$

where we take $T_s$ to be the temperature measured at the buried sensor closest to the surface, and the integral function is

$$
\Phi_{2h}(z) \equiv \int_{\xi=z_0}^{z} \phi_h d \ln \xi. \qquad (\text{D5})
$$

From Equation D1, the dimensionless heat flux is then

$$
q^* = \frac{\kappa v_\tau^* p^*}{\text{Pr}_t \Phi_{2h} (z_T^*/L^*)} \ln \left( \frac{T_T^*}{T_s^*} \right). \quad (\text{D6})
$$

Meanwhile, the Monin-Obukhov length in Equation 55, dimensionless with $(\alpha J)^{1/2}$, satisfies the equation

$$
L^* = \left( \frac{\alpha}{J^3 g^2} \right)^{1/2} \frac{v_{\tau}^{*2} \text{Pr}_t \Phi_{2h} (z_T^*/L^*)}{\kappa^2 \ln (T_T^*/T_s^*)}, \quad (\text{D7})
$$

which we solve for $L^*$ using time-histories of the measured wind speed $U$, temperatures $T_T$, and $T_s$ reported in Section 4, $v_\tau/U \approx 0.036$ measured at the probe location in January 2017, and adopting $\text{Pr}_t \approx 1$.

Appendix E: Soret Coefficient

Chapman and Cowling (1953) used the kinetic theory to derive the Soret diffusion mass flux $-\rho(D/\omega)k_T \nabla \ln T$ of a gas driven by a temperature gradient. They found
Meanwhile, transport in the ABL faces instabilities arising from natural convection during the day (Wyngaard, 2010), and it is complicated at night by stratified, weak turbulence (Mahrt, 2014; Optis et al., 2014) and by low-level jets intermittently perturbing the stable nocturnal boundary layer (Banta et al., 2007; Klein et al., 2016). For flat expanses of land, the resulting transports of momentum and heat are typically modeled using the Monin-Obukhov similarity (Monin, 1970). However, in the case of isolated sand dunes surrounded by a hard, more impermeable desert floor, it is unclear how such similarity can be extended to mass transfer of water vapor, or whether a boundary layer of vapor mass fraction developing from the dune leading edge may dominate the exchange. Therefore it is opportune to validate mass transfer boundary conditions at the base of the ABL at a realistic field scale with direct flux measurements from within the dune.

In this context, the capacitance probe described in section 2 provides an alternative to traditional evaporation instruments by capturing spatio-temporal variations of the water vapor mass fraction Y just below the sand surface. As section 8 showed, an integration of the governing equations then returns a best estimate of the dimensionless instantaneous wind-driven seepage velocity $u^*$. At the surface, denoted by the subscript s, this yields the superficial gas velocity

$$v_s^* = +(1 - \nu)u_s^* \qquad (53)$$

counted > 0 into sand from the atmospheric boundary layer (ABL).

In addition, by virtue of their accuracy and low noise, the recorded depth profiles of Y can be smoothed with a spline and differentiated to evaluate the vapor flux $\dot{\Psi}_s'' = -\rho(1 - \nu)(D/\bar{\omega})(\partial Y/\partial x)_s$ across the surface, again counted > 0 into sand. Made dimensionless with $\rho_{st}\sqrt{\alpha/J}$, the diffusive flux in Eq. (32) augmented by its advection counterpart $u_s^*Y_s p^*/T_s^*$, constitutes a unique measurement of the mass transfer boundary condition at the base of ABL. As Appendix H shows from the integral equations of the boundary layer, the proper way is to count it > 0 into the ABL aloft as

$$\Sigma^* = -\Psi_s^* - \frac{p^*}{T_s^*}v_s^*(Y_s - Y_a). \qquad (54)$$

In this determination, the principal uncertainty is associated with the tortuosity $\bar{\omega} \sim \pi/2$ (Shen & Chen, 2007), which we estimated for the nearly spherical and monodisperse grains of this dune.

As the panels of Figure 10 show, the diffusion flux into the ABL and its sign are, as expected, nearly proportional to the “driving potential” ($Y_s - Y_a$) in the second day, i.e. to the difference between the surface mass fraction $Y_s$ and its ambient counterpart $Y_a$. However, two surprising observations stand out. First, the driving potential nearly vanishes during aeolian transport in the first day (Fig. 10A), while the flux maintains roughly the same magnitude (Fig. 10B). As discussed in section 12, we attribute this phenomenon to enhanced mass transfer by saltating particles, which effectively produce a Dirichlet boundary condition imposing $Y_s \simeq Y_a$ at the surface.

The second paradox is more puzzling. When the ABL is stable at night, the dry surface keeps losing moisture to the wetter ambient, in an apparent inversion of the proportionality between flux and driving potential. In addition, the net surface flux changes sign in the last quarter of the night, while the driving potential does not.

Profile measurements at our short weather station are not sufficient to resolve this paradox. However, data support the following comments. As Fig. 3C showed, the ABL is stable at night, when a colder dune at $\sim 290$°K gives up heat at long-wavelength infrared $\sim 10$ µm, and unstable during the day, when warm air rises from sand stricken by a solar irradiation that exceeds wind-driven losses. Therefore, turbulent momentum in the ABL is coupled with natural convection during the day (Wyngaard, 2010). To capture this effect, Monin and Obukhov proposed a similarity between the transports of mo-
$$k_T = 5(C_s - 1) \frac{S_1[n_1/(n_1+n_2)] - S_2[n_2/(n_1+n_2)]}{Q_1(n_1/n_2) + Q_2(n_2/n_1) + Q_{12}}, \quad (E1)$$

where $n_1m_1$ is the partial density of gas 1 with molecular mass $m_1$ diffusing into gas 2. Substituting 1 for 2 and vice-versa completes the set of equations. With water vapor (gas 1) dilute in air (gas 2), $n_1m_1 = \rho Y$ and $n_2m_2 = (1 - Y)\rho \approx \rho$. Using the shorthand $M_1 \equiv m_1(m_1 + m_2)$, Chapman and Cowling (1953) calculated

$$S_2 = M_2^2 E_2 - 3M_1(M_1 - M_2) - 4M_2 M_1 A_s$$
$$Q_2 = M_2 E_2 [6M_1^2 + (5 - 4B_s) M_2^2 + 8M_2 M_1 A_s]$$
$$Q_{12} = 3(M_1 - M_2)^2 (5 - 4B_s) + 4M_1 M_2 A_s (11 - 4B_s) + 2M_1 M_2 E_1 E_2$$
$$E_2 = \frac{2\sqrt{2}}{5M_2 M_1^{1/2}} \left( \frac{\sigma_2}{\sigma_{12}} \right)^2$$
$$\sigma_{12} = (\sigma_1 + \sigma_2)/2,$$

$$E_2 = \frac{2\sqrt{2}}{5M_2 M_1^{1/2}} \left( \frac{\sigma_2}{\sigma_{12}} \right)^2$$
$$\sigma_{12} = (\sigma_1 + \sigma_2)/2,$$

with $A_s = 2/5$, $B_s = 3/5$, and $C_s = 6/5$. Because $n_1/n_2 \approx M^*Y \ll 1$, where $M^*$ is the ratio of the molar masses of air and water, Equation E1 simplifies to $k_T \approx -S_2M^*Y/Q_2 \equiv a_T Y$. With $M_1 = 1/(1 + M^*)$, $M_2/M_1 = M^*$, $\sigma_1 \approx 2.65$ Å, $\sigma_2 \approx 3.60$ Å, we find $a_T \approx -0.319$.

With this magnitude, our calculations indicate that log derivatives of temperature $\partial \ln T^*/\partial x^*$ and $\partial \ln T^*/\partial t^*$ are too small for terms involving $a_T$ to play a significant role in Equation 49. Therefore in our case, the Soret effect can be ignored. In extra-terrestrial applications with stronger temperature variations, this may no longer be the case.

## Appendix F: Seepage Profiles

On time scales long enough to establish equilibrium, the mass conservation Equation 15 and the rate Equation 24 can be manipulated to extract an ODE for the evolution of $u^*$ along the relative depth $x^*$,

$$\frac{\partial u^*}{\partial x^*} = \left[ -\frac{\partial \ln p^*}{\partial t^*} + \frac{\partial \ln T^*}{\partial t^*} + \frac{(M^* - 1)}{1 + Y(M^* - 1)} \frac{\partial Y}{\partial t^*} \right] \\ - u^* \left[ \frac{\partial \ln p^*}{\partial x^*} - \frac{\partial \ln T^*}{\partial x^*} - \frac{(M^* - 1)}{1 + Y(M^* - 1)} \frac{\partial Y}{\partial x^*} \right] \\ - \mathbb{R}\Omega_1 \frac{T^*}{p^*} \left[ f_T \frac{\partial T^*}{\partial t^*} + f_Y \frac{\partial Y}{\partial t^*} \right], \quad (F1)$$

which is coupled with Darcy's law (17) through the dimensionless term

$$\frac{\partial \ln p^*}{\partial x^*} = - \left( \frac{\mu \alpha}{K p_{st}} \right) \frac{u^*}{p^*}. \quad (F2)$$

These equations are solved with ODE (30), or with PDE (26). They are used to calculate profiles of $u^*$ in the insets of Figure 7.

## Appendix G: Kinetic Limit

In this appendix, we estimate the time $\tau_d$ needed to bring the mass fraction $\Omega_d$ held by a spherical grain of diameter $d$ to its value $\Omega_{e,d}$ at equilibrium with the imposed $Y$ around it. This equilibration process is driven by the volumetric rate of evaporation at the particle scale, which is governed by the surface mass flux $\dot{W}_{s}''$ expanding into the interstitial space of density $\rho(1 - \nu)$ surrounding grains. In Equation 44, $Y_e$ is the mass fraction at equilibrium with the instantaneous $\Omega_d$ on the solid of size class $d$ given by Equation B1.

As in the combustion of a solid particle (Kanury, 1975), there are two limiting regimes. In the “kinetic limit,” $k_m$ is small relative to $\rho D/d$, so diffusion erases gradients of $Y$, and the flux is given by Equation 43, $\dot{W}_{s}'' = \dot{W}_{HKS}''$. Combining with Equation 44, the characteristic time of size class $d$ is
<figure><img src="image_11.png" /></figure>

**Figure 10.** Time-histories in the atmospheric boundary layer (ABL). (A) Driving potential ($Y_s - Y_a$), nearly vanishing during aeolian transport from $t^* \simeq 1.4$ to 1.6, thus imposing a Dirichlet boundary condition on the ABL. (B) Thick line: measured dimensionless net flux of Eq. (54). Thin line: Turbulent boundary layer (TBL) model of Eq. (57) with $a_m \simeq 0.0010$ and $a_m \simeq 0.0015$ under stable ($L > 0$) and unstable ($L < 0$) conditions, respectively, $\Sigma^*_\infty \simeq 0.43$ in the first night, $\Sigma^*_\infty \simeq 0.22$ in the second night, and $\Sigma^*_\infty \simeq 0.075$ in the second day. Dotted line: TBL model of the second day extrapolated to the period of stability after sunrise. (C) Inverse Monin-Obukhov length, dimensionless with $\sqrt{\alpha J}$; dotted lines at $\pm$ the inverse dimensionless altitude $z_T$ of the top RH sensor. (D) $Re_X$ and $Gr_L^{1/2}$. The horizontal dashed line marks $Re_{X\text{crit}}$ (Kays & Crawford, 1980).
$$ \tau_d = \tau_{\text{HKS}} = \frac{\rho(1-\nu)d}{6\nu k_m} = \left(\frac{1-\nu}{\nu}\right) d \sqrt{\frac{\pi MW_{\text{H}_2\text{O}}}{18RT}} \left(\frac{1-\kappa_e/2}{\kappa_e}\right), \quad (G1) $$

which, upon averaging over the particle size distribution, would return the same expression with the Sauter diameter $d_{2,3}$ instead of $d$.

If instead diffusion is not as fast, it is essential to involve it in the local water mass balance, as the local vapor mass fraction $\tilde{Y}$ may vary within each pore. For simplicity, we imagine, crudely, that particles are isolated, as commonly assumed to derive closed-form solutions for solid or droplet combustion (Kanury, 1975). Under quasi-steady conditions $\partial \tilde{Y} / \partial t \approx 0$, the mass flow rate through any spherical shell at radius $r$ from the particle center is invariant and equal to its value $\pi d^2 \dot{W}_s''$ at the surface. Then, the isothermal balance of water vapor mass in a slice $[r, r + dr]$ is

$$ \pi d^2 \dot{W}_s'' \frac{\partial \tilde{Y}}{\partial r} - \rho D \frac{\partial}{\partial r} \left( r^2 \frac{\partial \tilde{Y}}{\partial r} \right) = 0. \quad (G2) $$

At $r = d/2$, the net evaporation rate of a unit area of water film must balance the sum of the advection and diffusion fluxes in the gas phase,

$$ \dot{W}_s'' = \dot{W}_s'' Y_s - \rho D \left( \frac{\partial \tilde{Y}}{\partial r} \right)_s. \quad (G3) $$

Integrating Equation G2 twice subject to $\tilde{Y} \to Y$ at $r \to \infty$ and boundary condition (Equation G3), we find

$$ (1 - \tilde{Y}) = (1 - Y) \exp[-d^2 \dot{W}_s''/(4\rho D r)], \quad (G4) $$

where $\dot{W}_s''$ is given by Equation 43. Applying this profile expression to the surface at $r = d/2$ and eliminating $Y_s$ using Equation 43 yields an equation to solve for $\dot{W}_s''$ in the general case,

$$ 1 - Y_e + \frac{\dot{W}_s''}{k_m} = (1 - Y) \exp \left( - \frac{d \dot{W}_s''}{2\rho D} \right). \quad (G5) $$

As expected, we recover $\dot{W}_s'' = \dot{W}_{\text{HKS}}''$ in the kinetic limit by making $D \to \infty$. In the “diffusion limit,” slow diffusion is the rate limiting step. Here, by making $k_m \to \infty$, $\dot{W}_s''$ becomes

$$ \dot{W}_{\text{DIFF}}'' = \frac{2\rho D}{d} \ln \left( \frac{1 - Y}{1 - Y_e} \right) \approx \frac{2\rho D}{d} (Y_e - Y), \quad (G6) $$

with characteristic time

$$ \tau_{\text{DIFF}} = (1 - \nu)d^2/(12\nu D). \quad (G7) $$

Because an interstitial space with $\nu > 0.5$ has a pore size on the order of $d[(1 - \nu)/\nu]^{1/3} < d$, the time to diffuse vapor through this space should be even less than predicted by Equation G7. Combining the profile Equation G4 with Equation G6, we also find $Y_s = Y_e$ in the diffusion limit.

Expanding the exponential to first order in Equation G5 and recognizing that $Y \ll 1$, the flux in the general case is approximately given by

$$ \frac{1}{\dot{W}_s''} \approx \frac{1}{\dot{W}_{\text{DIFF}}''} + \frac{1}{\dot{W}_{\text{HKS}}''}, \quad (G8) $$

confirming that the two limiting processes operate in parallel. Damköhler's “second ratio” $Da_{II} \equiv \tau_{\text{DIFF}}/\tau_{\text{HKS}} = k_m d/(2\rho D)$ reveals which mechanism predominates by being the slowest. A large $Da_{II}$ implies the diffusion limit. With a unity accommodation coefficient of evaporation ($\kappa_e \sim 1$), this would likely be the case, since $Da_{II} \sim 860$ for our mean Sauter diameter under standard conditions. For example, to complement the capillary condensation model of Philip (1964) and Shahraeeni and Or (2010) postulated a diffusion-limited process for film adsorption at the microscopic scale of wedge-shaped grain asperities. However, as Section 12 shows, the characteristic time
<figure><img src="image_12.png" /><figcaption>**Figure 11.** (A) Small difference $\Delta Y_a = Y_a(z_T) - Y_a(z_B)$ in the ambient water vapor mass fraction time-history between the top and bottom sensor elevations at $z_T \simeq 2$ m and $z_B \simeq 0.9$ m measured by the weather station above hard ground with much higher aerodynamic roughness $z_0 \simeq 0.011$ m (Louge et al., 2013) than found over the sand surface. Vertical lines mark sunrise and sunset. The horizontal dashed line is the mean $\Delta Y_a$. (B) Corresponding change in ambient temperature $\Delta T = T(z_T) - T(z_B) \equiv T_T - T_B$ (°K). (C) Ratio $U(z_T)/U(z_B)$. The horizontal dashed and dotted lines show $\ln(z_T/z_0)/\ln(z_B/z_0)$ and $z_T/z_B$, respectively consistent with a turbulent log-law and a viscous sublayer.</figcaption></figure>
involved $\tau_{\text{DIFF}} \sim 240$ µs would be much too small to explain the exponential decay of $\Delta Y$ with depth. Therefore, the behavior of subsurface waves imply instead that the kinetic limit prevails at the grain scale.

## Appendix H: Developing Mass Transfer Boundary Layer

We invoke integral equations of the boundary layer to model wind-driven convective mass transfer, coupled to the advection/diffusion measured through the sand surface, with possible mass flux exchanged with the ABL aloft. Unlike Appendix D, where we entertained unsteady terms, here we ignore them due to the more rapid development of the turbulent mass transfer boundary layer (TBL). Overall mass conservation in a TBL slice perpendicular to the surface is

$$ \frac{d}{d\mathcal{X}} \int_{z=0}^{z_{\ell}} \rho U dz - \rho_{\ell} v_{\ell} + \rho_s v_s = 0, \qquad (H1) $$

where $\mathcal{X}$ is distance along the wind from the leading toe of the dune, $v$ is superficial gas velocity (>0 toward sand), and indices $s$ and $\ell$ represent the surface and the top of the slice, respectively. The corresponding mass balance of water vapor is

$$ \frac{d}{d\mathcal{X}} \int_{z=0}^{z_{\ell}} \rho U Y dz - \rho_{\ell} Y_{\ell} v_{\ell} + \rho_s Y_s v_s - \dot{\Psi}_{\ell}'' + \dot{\Psi}_s'' = 0, \qquad (H2) $$

where vapor fluxes $\dot{\Psi}''$ are >0 into sand. Eliminating $\rho_{\ell} v_{\ell}$ from Equation H2 using Equation H1, dividing by $\rho_{\ell} U_{\ell}$ and letting $\ell \to \infty$, we find

$$ \frac{d\delta_m}{d\mathcal{X}} + \frac{\rho_s v_s}{\rho_\infty U_\infty} - \frac{\dot{\Psi}_\infty''}{\rho_\infty U_\infty (Y_s - Y_\infty)} + \frac{\dot{\Psi}_s''}{\rho_\infty U_\infty (Y_s - Y_\infty)} = 0, \qquad (H3) $$

where $\rho_\infty$ and $U_\infty$ are recorded at an altitude higher than the mass transfer thickness of the boundary layer

$$ \delta_m \equiv \int_{z=0}^{\infty} \frac{\rho U}{\rho_\infty U_\infty} \left( \frac{Y - Y_\infty}{Y_s - Y_\infty} \right) dz. \qquad (H4) $$

Because water vapor has a diffusion coefficient $D$ on the same order as the thermal diffusivity of air or, equivalently, that its Lewis number is near unity, an analogy with heat transfer in the TBL suggests that $d\delta_m/d\mathcal{X}$ is a kind of Stanton number

$$ \text{St}_X = \frac{d\delta_m}{d\mathcal{X}} \approx \frac{a_m}{\text{Re}_X^{1/5} \text{Sc}^{0.4}}, \qquad (H5) $$

where $\text{Sc} \equiv (\mu/\rho)/D \simeq 0.60$ is the Schmidt number of water vapor diffusing in air. Then, from its standpoint, the ABL receives a net surface flux

$$ \dot{\Sigma}'' \equiv -\dot{\Psi}_s'' - \rho_s v_s (Y_s - Y_a) = a_m \frac{\rho_\infty U (Y_s - Y_a)}{\text{Re}_X^{1/5} \text{Sc}^{0.4}} + \dot{\Sigma}_\infty'', \qquad (H6) $$

where $\dot{\Sigma}'' \equiv -\dot{\Psi}''$ is the upward moisture flux on top of the TBL, with both $\dot{\Sigma}''$ and $\dot{\Sigma}''_\infty$ now counted >0 upwards, and we identify $Y_\infty$ and $U_\infty$ as the mass fraction $Y_a$ and wind speed $U$ recorded by the top instruments of the weather station. Using this equation and our net flux measurements provided in dimensionless form in Equations 31–32 we estimate the coefficient $a_m$ and the exchange flux $\dot{\Sigma}''_\infty$ toward the ABL aloft (Section 13).

## Conflict of Interest

The authors declare no conflicts of interest relevant to this study.
mentum, heat and mass in the turbulent ABL (Monin, 1970), which we summarize in Appendix D. The similarity is captured by corrections to the Prandtl-von Kármán log-law of the wall using heuristic functions $\phi_m(z/L)$, $\phi_h(z/L)$ and $\phi_Y(z/L)$ for momentum, heat and moisture. In these functions, the altitude $z$ is measured against the Monin-Obukhov length

$$L = \frac{(\rho c_p T) v_{\tau}^3}{\kappa g \dot{q}_{\text{wind}}''}, \qquad (55)$$

which is positive when the cold dune is warmed by a stable ABL at night ($\dot{q}_{\text{wind}}'' > 0$), and negative when an unstable ABL cools off sand during the day. In Eq. (55), $g$ is the gravitational acceleration, $\kappa \approx 0.41$ is von Kármán's constant, $T$ is absolute temperature, and $\dot{q}_{\text{wind}}''$ is the convective thermal flux at the surface. (Note that, because we count $\dot{q}_{\text{wind}}'' > 0$ into sand, the expression of $L$ in Eq. (55) has opposite sign than most conventions for the ABL). For example, field tests in Kansas in 1968 conformed to

$$\phi_m = \begin{cases} 1 + a_{m_N} z/L & , L > 0 \\ 1/(1 - a_{m_D} z/L)^{1/4} & , L < 0 \end{cases} \qquad (56)$$

with coefficients $a_{m_N} \approx 4.8$ and $a_{m_D} \approx 19.3$ (Wyngaard, 2010). At low altitudes such that $z \ll |L|$, all three functions tend to 1, and the log-law remains unchanged. Such is the case for our relatively short weather station (Fig. 10C).

The Monin-Obukhov analogy was meant for a fully-developed ABL established over vast, uniform, flat expanses, like the horizontal desert floor surrounding our sparsely-populated barchan dunes. However, it is questionable whether its mass transfer predictions apply to the dune itself during the day, or whether it is altogether valid under stratified, weak turbulence at night (Mahrt, 2014; Optis et al., 2014).

As Fig. 11A shows, ambient moisture changes negligibly with altitude at the weather station, $\Delta Y_a \approx -0.01 Y_a$, in contrast with the noticeable difference in ambient temperature at the same location (Fig. 11B). Therefore, the moisture flux through the hard, relatively more impermeable desert floor should be very small. However, our measurements reveal that it is significant above the porous sand surface (Fig. 10B), which typically traps more moisture during long periods without rain (Louge et al., 2013). Consequently, the windward toe of the dune likely constitutes the leading edge of a turbulent mass transfer boundary layer (TBL) driven by forced horizontal convection, developing below an ABL subject to vertical natural convection when unstable, and stratified otherwise.

We consider such boundary layer in Appendix H. To establish its Reynolds number $Re_X \equiv \rho U \chi / \mu$ at the probe, where $\chi \approx 1.8 \times 10^{-5}$ kg/m.s is the air dynamic viscosity, we calculate the distance $\chi$ from the toe to the buried probe along the instantaneous wind direction. As the dashed line in Fig. 10D shows, $Re_X$ almost always exceeds the critical value $Re_{X,crit} \approx 3 \times 10^5$ for transition to a TBL, except from sunset to midnight, when the lower ABL intermittently behaves as a viscous sublayer or as log-law turbulence (Fig. 11C).

To make the TBL analysis tractable, we invoke the integral equations of the boundary layer (Kays & Crawford, 1980), and adopt the analogy between turbulent heat and mass transfer to estimate a Stanton number capturing wind-driven mass transfer. We account for exchanges at the upper boundary of the TBL by including a vapor flux $\Sigma''_{\infty}$ counted > 0 into the ABL aloft. Equation (H6) then predicts the net dimensionless flux, which we equate to what we measured in Eq. (54),

$$\Sigma^* \equiv -\Psi_s^* - \frac{p^*}{T_s^*} v_s^* (Y_s - Y_a) = a_m \frac{\rho^* U^* (Y_s - Y_a)}{\text{Re}_X^{1/5} \text{Sc}^{0.4}} + \Sigma_{\infty}^*, \qquad (57)$$

where $\rho^* \equiv \rho / \rho_{st} \approx p^*/T_T^*$ is the relative air density in the TBL, and $\text{Sc} \equiv (\mu/\rho)/D \approx 0.60$ is the Schmidt number. As Fig. 10 shows, this prediction requires a value of $a_m$ about
## Data Availability Statement

The entire data set, movie animations, and nomenclature are available as Supporting Information at https://doi.org/10.7298/kqgg-5888. The file “LouveEtAl_JGR-ES_2022_ReadMe” lists their contents, including movies animating Figures 4, 6 and 8, respectively called Movie S1, S2 and S3, and it specifies all parameters adopted in this work.

## Acknowledgments

The authors are grateful to Patrick Perré, Floran Pierre, and Joël Casalinho for help measuring sand isotherms; to Jean-Luc Métayer for measuring particle size distribution, to Anthony Hay, Ali Sultan, Renée Richer, Christopher Ogden, Dah Ould Ahmedou, Pascal Dupont, Anushree Acharya, Sara Abdul-Majid, Osama Al Achek, Carlos Mejia, Nathalie Vriend, and Matthew Arran for assistance in the field; to Chun-Chieh (Jimmy) Chang for creating special batteries; to Robert Foster and Hai Trinh for advice on capacitance instrumentation; to Stephen Keast for help with probe design; to Marc Massot, Greg Bewley, Abraham Stroock, Jean-Yves Parlange, Tammo Steenhuis, Shmuel Assouline, Dani Or, Michael Gentzler and Sylvain Dupont for illuminating discussions; to Isabelle Louge and J. Gregory Butler for vetting the plain language summary; to Wendy Kozlowski for curating the data set; and to Amy East, Thomas Pähtz, Giora Kidron and anonymous reviewers for their insight on revising the manuscript. This paper was made possible by the support of NPRP grants 09-546-2-206 and 6-059-2-023 from the Qatar National Research Fund, and a Qatar Foundation Research Excellence Award.

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an order of magnitude smaller than a typical laboratory correlation $a_m \simeq 0.0307$ for the TBL over a flat plate (Kays & Crawford, 1980), suggesting a much thicker TBL than expected. However, this disagreement may not be surprising, since Reynolds et al. (1958) fixed $a_m$ for heat transfer (not mass transfer) at Reynolds numbers within $9.2 \times 10^4 < \text{Re}_X < 1.27 \times 10^6$, which are below values in Fig. 10D, except during quiet periods at night. Another possible reason for the discrepancy may be a possible gradual evolution of $Y_s$ along the dune, which we did not measure. Lastly, to gauge whether the apparent TBL thickening may be associated with the presence of strong natural convection during the day, Fig. 10D also compares $\text{Re}_X$ with the square root of a Grashof number $\text{Gr}_L^{1/2} \equiv \rho[gL(T_s - T_T)/T_T]^{1/2}L/\mu$, which amounts to a Reynolds number based on $L$ and speed driven by buoyant air rising from the hot surface at $T_s$ and the colder measured ambient temperature $T_T$. However, because $\text{Gr}_L^{1/2} < \text{Re}_X$ during much of the day, natural convection is unlikely to dominate mass transfer in the TBL.

At night, fitting our data to Eq. (57) also suggests a strong upward vapor flux $\Sigma_{\infty}^*$ at the top edge of the TBL, perhaps due to a moisture inversion above the dune, whereby dry air aloft lurks above a more humid layer closer to the surface (Mahrt, 2014). However, because the turbulent behavior of the stable night-time ABL is complex (Optis et al., 2014; Mortarini et al., 2016), we do not know whether this interpretation has merit. In future, detailed profiles of $Y_s$, $T$ and $U$ vs elevation should be simultaneously established above a capacitance instrument recording surface moisture flux to inform the detailed behavior of the mass transfer ABL, especially at night.

## 14 Conclusions

In this work, we deployed a unique capacitance instrument of fine vertical resolution to measure spatio-temporal variations of water mass fraction adsorbed on sand grains in the hygroscopic regime without free liquid, as well as stratigraphy of the solid volume fraction, within the first 30 cm from the surface of a hyper-arid dune. We simultaneously recorded temperature profiles there, as well as ambient conditions, net solar radiation, and wind speed and direction. We also characterized relevant sand properties, including the van der Waals isotherm that relates water mass fraction in the bulk solid to relative humidity around grains at equilibrium.

Unlike heat, which propagates mainly through the sand contact network by conduction, water vapor diffuses and advects through the tortuous interstitial pore space between grains. As a result, its governing equations are more complicated and lead to richer physics. Accordingly, we derived a partial differential equation (PDE) describing the unsteady advection-diffusion of water vapor in that space without free liquid, coupled with an ordinary differential equation (ODE) capturing evaporation kinetics of water adsorbed on solid grains.

We used this framework to interpret data. First, we considered long diurnal time scales, during which grains have ample opportunity to achieve thermodynamic equilibrium with the surrounding relative humidity. Here, the source term in the solid phase ODE effectively augments the unsteady term of the water vapor PDE. The result is an apparent vapor diffusion that is nearly three orders of magnitude slower than without any solid. In simpler terms, each dry sand layer hinders vapor diffusion by adsorbing moisture before letting it proceed to the next layer.

Another attribute of the long-term equilibrium behavior is a strong coupling between depth profiles of water vapor mass fraction and temperature. As the latter evolves on a diurnal basis, the non-linear coupling produces multiple inflections in moisture mass fraction that have no counterpart in linear diffusive systems. It originates in the dependence of the amount of adsorbed water on interstitial relative humidity, which varies exponentially with temperature through the saturation pressure. We considered two other
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transport processes driven by strong temperature gradients, namely the Soret effect and
natural convection. However in our case, both proved negligible.

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The porous matrix created by well-rounded packed sand also allows a slow “seep-
age” advection to compete with diffusion, or to augment it. To evaluate its velocity at
each measurement time step, we transformed the coupled governing equations into a sin-
gle ODE along depth, which we solved as a two-point boundary value problem subject
to the measured vapor mass fraction just below the surface and at the deepest probe sen-
sor that consistently yielded a stable signal. We then selected the seepage velocity that
presented the smallest least-squares error between ODE prediction and data.

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Separately, we showed that seepage is caused by gentle variations in dune topography that induce streamwise surface static pressure oscillations decaying exponentially with depth, thereby creating Darcy pressure gradients that effectively amplify small topographical deformations. As such, seepage changes from vapor inhalation to exhalation along the surface, and it depends on wind direction.

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Then, we applied the governing equations to the other limit where water mass fraction varies on short time scales. This let us elucidate a peculiar moisture wave phenomenon, whereby disturbances in vapor mass fraction on the dune surface propagate rapidly downward into its sands. A linear stability analysis of the governing equations identified the origin of these evanescent waves. Whereas subsurface moisture transport on the decimetric scale operated in an effective diffusion limit, we showed that the microscopic size of the interstitial space between grains was too small for the evaporation process itself to be diffusion-limited. Accordingly, we derived an estimate for the characteristic time of first-order drying kinetics from the exponential decay of the moisture wave amplitude with depth. Here, data suggested a drying process at the grain scale that is limited by the kinetics of a reaction with activation energy larger than, but on the order of, the latent heat of vaporization.

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Finally, the depth profiles of water mass fraction Y allowed us to determine the time-
history of the moisture flux into the atmospheric boundary layer (ABL). To our knowl-
edge, this constitutes the first direct measurement of this quantity on a hyper-arid sur-
face. We registered three unexpected observations. First, when the flux was, as antic-
pated, nearly proportional to the difference (Y_s − Y_a) between vapor mass fraction at
the surface and in the ambient, it was about an order of magnitude smaller than pre-
dicted for a turbulent mass transfer boundary layer developing from the dune toe, or by
a Monin-Obukhov correlation of the ABL. Second, during aeolian sand transport, the
flux was effectively decoupled from a vanishing (Y_s − Y_a). Third and most intriguing,
although the flux slowly decreased in unisson with (Y_s − Y_a) during the stable ABL at
night, it was augmented by a nearly invariant upward moisture flux aloft. To elucidate
these observations in future, detailed profiles of vapor mass fraction Y, wind speed U
and temperature T in the ABL should be recorded just above probes measuring Y and
T in the subsurface.

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The capacitance instrument and modeling framework that we presented in this art-
ticle have wider relevance than hyper-arid dunes. In industry, they can be deployed, for
example, to predict vapor contamination through pharmaceutical powders (Louge et al.,
2021). Because the probe can reliably detect very low moisture, it could also be used as
“ground-truth” for satellite remote sensing over sand seas (Zribi et al., 2014; Myeni et
al., 2019; Bürgi & Lohman, 2021), or to look for scant water in future space applications (Honniball
et al., 2020; Davidsson & Hosseini, 2021).

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albeit on a relatively coarse spatial resolution. There, the principal challenge is to specify the water retention curve, a history-dependent relation between water volume fraction and capillary pressure subject to “return-point” memory hysteresis, which is predictable for known pore networks (Xu & Louge, 2015), but is typically modeled instead with the Van Genuchten (1980) empirical correlation, see for example Assouline and Kaimai (2019), or with local closures involving meniscus curvature (Gray & Miller, 2005) that elicit controversy (Baveye, 2013).

For hyper-arid soils without free liquid, measurements of small amounts of water adsorbed on solids lacked until now the spatial resolution and accuracy that could reveal the essential physics of water transport. With such instrument at hand, we have shown that a model free of arbitrary closures can capture observations and record the exchange of moisture at the soil-atmosphere interface. Recent improvements in the stability of the instrument's processing electronics suggest that its range can be extended to greater water content (Louge et al., 2021), thereby enabling measurements into the more humid vadose zone with a single instrument.

## Appendix A Multiple capacitance probe

Louge et al. (1997) developed capacitance instruments to measure density and dielectric signature of snow. Recently, Louge et al. (2021) improved their data reduction to reach unprecedented accuracy in pharmaceutical applications. Our design is based on a similar principle (Fig. 1). The probe records the impedance of the medium between any one of 15 sensors and a “target” held at a constant reference voltage. To achieve this, conductors carrying a sensor voltage are surrounded by a “guard” driven at precisely the same potential by an independent circuit. By guiding electric field lines that penetrate sand at the probe face, the guard focuses the extent of the probe’s measurement volume originating from the sensor, thereby avoiding external interference known as “stray” capacitance. Because the guard is also connected to the outer conductor of a high-quality coaxial cable connecting probe and processing electronics, it also shields the wire and all electrical components carrying the sensor voltage. Therefore, because there can be no charge accumulation in the cable, the latter’s capacitance does not perturb the measurement, unlike conventional bridge circuits. Consequently, this technique allows precise detection of extremely small capacitances.

For the invasive design in Fig. 1, symmetry produces circular electric field lines that are shed from sensor to target in planes perpendicular to the long axis of the probe. Their center lies on the target conductor at a distance $x_c$ from the centerline and with radius $R$ satisfying $x_c^2 - R^2 = a^2$, where $a \approx 12.6$ mm is the half guard width (Louge et al., 1996). These field lines delimit a measurement volume of $W \approx 5$ mm vertical spatial resolution set by the sensor height $W$, small enough to resolve steep gradients of $\Omega$ perpendicular to the free surface. In the probe of Fig. 1, each sensor has respective outer and inner distances $b \approx 11.8$ mm and $c \approx 4.8$ mm from the lance’s centerline. It is separated from the grounded target by a thin guarded strip that absorbs the singular voltage jump from AC guard to DC target. As Louge et al. (1996) calculated, this makes the measurement volume penetrate sand no farther than $(a^2 - c^2)/2c \approx 14$ mm from the probe. It also confers each sensor a capacitance $C_0 = \epsilon_0 l_c = \epsilon_0(W/\pi) \ln\{(b+a)(c-a)/[(b-a)(c+a)]\} \approx 38 \ 10^{-15}$ F when the probe is exposed to air or, equivalently, $l_c \approx 4.3$ mm.

Such tight dimensions are achieved by deploying all conductive surfaces in a thin printed-circuit board (PCB) consisting of four layers. The top one is where sensor, guard and target are exposed to sand. To limit wear, it is covered with a thin TAIYO PSR-4000BN solder mask. The next layer is a copper film held at guard voltage, designed to shield the back of all sensors. The third layer contains wires of resistance $\lesssim 1 \Omega$ bringing sensor voltages to a location where each can be individually soldered to one of 15 wires as-
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embodied into a guarded coaxial cable ~ 1.5 m linking probe and processing electronics. Finally, the bottom PCB layer is another guarded copper film shielding these sensor wires. Our PCB was printed by ADVANCED CIRCUITS.

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Coincidentally, the thin solder mask, which equilibrates quickly to the local inter-
stitial vapor mass fraction Y, also lets the probe discern rapid changes in that quantity, even if sand lags in achieving an adsorbed mass fraction Ωₑ at equilibrium with Y. Al-
though Ω is the primary variable determining the recorded complex effective dielectric
constant, adsorbed moisture on the film, which is closest to sensor and ground, can af-
fect the electric field, thereby acting as an impedance in series with the sand's, regard-
less whether Ω has reached Ωₑ. We noticed this behavior when the phase lead of the bare
probe in moist air disappeared after it was briefly dried in an oven. While it is unclear
whether the probe detects rapid variations in Y quantitatively, we exploited such capa-
bility to reveal the signal bistability shown in Fig. 3, which we analyzed in section 12.

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The electronics, manufactured by CAPACITEC, supplies sensors with a current of constant amplitude by controlling the AC voltage across a reference impedance fed with a stable “clock” oscillator of frequency f ≃ 15.6254 kHz, thereby producing sensor and guard voltage amplitudes proportional to the impedance between sensor and ground (Louge et al., 1997). When the probe is exposed to air alone, its impedance Z₀ = (2πfC₀)⁻¹ is only function of the capacitance C₀ between sensor and target. In humid sand, the impedance becomes Z = [2πfC₀(K''_e+ιK'_e)]⁻¹. By forming the ratio of guard voltage amplitude in air V₀ and in sand V, the probe then records the modulus

V₀/V = (K''_e + K''''_e)¹/². Louge et al. (2021) offer algorithms to determine guard amplitude and phase with high accuracy. Using them, the phase yields tan φ ≡ K''_e/K'_e and, ultimately, as shown in section 4, the acquired clock and guard signals provide sand bulk density and Ω simultaneously at each sensor (Louge, Valance, Babah, et al., 2010).

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The PCB is backed by a rigid composite lance made of DELRIN<sup>TM</sup> plastic and a glass filler, with effective heat conductivity ≃ 0.35 W/m·°K. This precaution minimizes thermal interference with surrounding sands of similar conductivity k<sub>s</sub> ≃ 0.49 W/m.K. (A metal probe, which we tried unsuccessfully, dried sand deeply by conducting heat from the surface at high noon). The hard-plastic housing lance is tapered to an asymmetric thin edge with 10° angle to impart minimum sand compression in front of the sensors during probe insertion.

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The top-rear of the lance supports a carefully-guarded enclosure to shield 15 sen-
sor wires connecting the PCB to a AMPHENOL multi-position circular connector plug.
The latter serves as quick connection with a guarded coaxial cable carrying all 15 sen-
sor wires to a guarded multiplexer. Upon receiving a five-bit binary address, the mul-
tiplexer operates relay switches to connect one of the sensors to the processing electron-
ics, while guarding all other. Relays and digital electronics in the multiplexer are pow-
ered by a small battery. To avoid stray capacitances within the multiplexer, the DC volt-
age of this battery is made to “float” with respect to the ground reference of the probe,
i.e. these two voltages are strictly independent.

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A compact Field Programmable Gate Array (FPGA) deployed in a NATIONAL IN-
STRUMENT cRIO controls the multiplexer by sending it a binary address through a NI-
9401 module, consistent with a pre-programmed schedule. The cRIO acquires 150 clock
and guard voltages simultaneously using two separate synchronous NI-9201 data acqui-
sition cards at a rate of 350 kS/s. It also acquires data from the Kipp & Zonen radiome-
ter and the temperature probe of Louge et al. (2013). Finally, the cRIO orchestrates data
storage on a USB-A memory flash drive. After retrieval from the latter, a MATLAB al-
gorithm described by Louge et al. (2021) calculates K'_e and K''_e. The system is powered
by a distant 12 V lead-acid battery recharged by a 1 kW solar panel, or by lithium-polymer
battery packs buried nearby for minimal footprint.
## Appendix B Size distribution

In the film adsorption regime, spherical sand particles of diameter $d$ and material density $\rho_p$ at equilibrium with the surrounding RH$_e$ have masses $(\pi/6)d^3\rho_p$ of solid and $\pi d^2\ell_w\rho_w$ of liquid, thereby holding a solid-bound water mass fraction

$$ \Omega_{d,e} = \frac{\Omega_{1_d}}{[T^* \ln(1/RH_e)]^{1/3}} \approx \frac{\Omega_{1_d}}{\{T^* \ln[p_{\text{sat}}/(pM^*Y_e)]\}^{1/3}}, \quad (B1) $$

where

$$ \Omega_{1_d} = \frac{6\ell_0 \rho_w}{d \rho_p} \quad (B2) $$

is inversely proportional to grain diameter (Shahraeeni & Or, 2010). In these expressions, the subscripts $e$ and $d$ indicate, respectively, equilibrium and the particle size class of diameter $d$. To average $\Omega_{1_d}$ over the entire population, we use the normalized particle size distribution $f_M$, such that the elementary mass fraction of grains with diameter in the range $[d, d + dd]$ is $f_M dd$. The result

$$ \Omega_1 = \int_{d=0}^{\infty} \Omega_{1_d} f_M dd = 6\ell_0 \frac{\rho_w}{\rho_p} \int_{d=0}^{\infty} f_M \frac{dd}{d} = \frac{6\ell_0}{d_{2,3}} \frac{\rho_w}{\rho_p}, \quad (B3) $$

which appears in the isotherm of Eq. (3), involves the “Sauter” diameter $d_{2,3}$. It is obtained by relating $f_M = d^3 f_N / \int_0^\infty f_N d^3 dd$ to the normalized population size distribution $f_N$. The notation $d_{2,3}$ adopts the convention of Babinsky and Sojka (2002),

$$ d_{p,q} \equiv \left[ \frac{\int_{0}^{\infty} d^p f_N dd}{\int_{0}^{\infty} d^q f_N dd} \right]^{1/(p-q)}. \quad (B4) $$

Louve et al. (2013) provided moments of the measured distribution $f_M$, summarized in their supporting information.

## Appendix C Isotherm derivatives

We define

$$ \begin{aligned} \frac{\partial \Omega_e}{\partial t} &\equiv \Omega_1 \left[ \frac{f_T}{T_{\text{st}}} \frac{\partial T}{\partial t} + f_Y \frac{\partial Y}{\partial t} + \frac{f_p}{p_{\text{st}}} \frac{\partial p}{\partial t} \right] \quad (C1) \\ &= \left[ \left( \frac{\partial \Omega_e}{\partial T} \right)_{\text{RH}} + \left( \frac{\partial \Omega_e}{\partial \text{RH}} \right)_T \times \left( \frac{\partial \text{RH}}{\partial T} \right)_{Y,p} \right] \frac{\partial T}{\partial t} \\ &\quad + \left[ \left( \frac{\partial \Omega_e}{\partial \text{RH}} \right)_T \times \left( \frac{\partial \text{RH}}{\partial Y} \right)_{p,T} \right] \frac{\partial Y}{\partial t} \\ &\quad + \left[ \left( \frac{\partial \Omega_e}{\partial \text{RH}} \right)_T \times \left( \frac{\partial \text{RH}}{\partial p} \right)_{T,Y} \right] \frac{\partial p}{\partial t}, \end{aligned} $$

where subscripts indicate which variables are held constant in calculating the derivative. Differentiating Eq. (3) and Eqs. (6)-(7), we find

$$ \begin{aligned} \left( \frac{\partial \text{RH}}{\partial Y} \right)_{p,T} &= \left( \frac{p}{p_{\text{sat}}} \right) \frac{M^*}{[1 + Y(M^* - 1)]^2}, \quad (C2) \\ \left( \frac{\partial \text{RH}}{\partial T} \right)_{Y,p} &= -\text{RH} \frac{T_{A_o}}{(T - T_{A_o})^2}, \\ \left( \frac{\partial \text{RH}}{\partial p} \right)_{T,Y} &= \frac{M^*}{p_{\text{sat}}} \times \frac{Y}{1 + Y(M^* - 1)}, \end{aligned} $$
and

$$ \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial T} \right)_{\text{RH}} = - \frac{1}{3T_{\text{st}} \ln^{1/3}(1/RH)} \left( \frac{T_{\text{st}}}{T} \right)^{4/3}, \qquad (\text{C3}) $$

$$ \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial \text{RH}} \right)_T = \frac{1}{3\text{RH} \ln^{4/3}(1/\text{RH})} \left( \frac{T_{\text{st}}}{T} \right)^{1/3}, $$

which can be substituted into Eq. (C1) to find expressions for the dimensionless functions $f_T$, $f_Y$ and $f_p$ in terms of $Y$, $T$ and $p$. In particular,

$$ \begin{gathered} f_Y \equiv \frac{1}{\Omega_1} \left( \frac{\partial \Omega_e}{\partial Y} \right)_{T,p} = \qquad (\text{C4}) \\ \frac{1/3}{Y [1 + Y(M^* - 1)] T^{*1/3} \ln^{4/3}(1/\text{RH})} \approx \frac{1/3}{Y T^{*1/3} \ln^{4/3} [p_{\text{sat}}/(pM^*Y)]}. \end{gathered} $$

Because grains of diameter $d$ have isotherms captured by Eq. (B1), they are individually subject to Eqs. (C3)-(C4), in which $\Omega_{1d}$ is substituted for its average $\Omega_1$ over the particle size distribution.

## Appendix D Monin-Obukhov similarity

In the spirit of Prandtl's mixing length hypothesis, Monin and Obukhov postulated that the turbulent transport of momentum is coupled to the fluxes of heat $\dot{q}''_{\text{wind}}$ and vapor mass $\dot{\Psi}''$ in the ABL through the similarity (Monin, 1970)

$$ \rho v_{\tau}^{2} = \frac{\kappa z \rho v_{\tau}}{\phi_{m}} \frac{\partial U}{\partial z}, \qquad (\text{D1}) $$

$$ \dot{q}''_{\text{wind}} = \frac{\kappa z \rho c_p v_{\tau}}{\phi_h \text{Pr}_t} \frac{\partial T}{\partial z}, $$

$$ \dot{\Psi}'' = \frac{\kappa z \rho v_{\tau}}{\phi_Y \text{Sc}_t} \frac{\partial Y_a}{\partial z}, $$

where $\kappa z$ is the mixing length, $c_p$ is specific heat at constant pressure, and $\text{Pr}_t$ and $\text{Sc}_t$ are, respectively, turbulent Prandtl and Schmidt numbers that Monin (1970) called “rattios of exchange coefficients” between heat and momentum or mass and momentum. Note that, for consistency with the internal flux, $\dot{\Psi}''$ and $\dot{q}''_{\text{wind}}$ in the ABL are positive downward. The quantities $\phi_m$, $\phi_h$ and $\phi_Y$ are heuristic functions of the relative altitude $z/L$ mentioned in section 13. For heat transfer, Louge et al. (2013) adopted

$$ \phi_h = \begin{cases} 1 + a_{h_N} z/L & , L > 0 \\ 1/(1 - a_{h_D} z/L)^{1/2} & , L < 0 \end{cases} \qquad (\text{D2}) $$

with coefficients $a_{h_N} \simeq 7.8$ and $a_{h_D} \simeq 12$.

Making $\rho$, $T$, $v_\tau$, $\dot{q}''_{\text{wind}}$ and $z$ dimensionless with $\rho_{\text{st}}$, $T_{\text{st}}$, $(\alpha/J)^{1/2}$, $\rho_{\text{st}}c_pT_{\text{st}}(\alpha/J)^{1/2}$ and $(\alpha J)^{1/2}$, respectively, the unsteady heat balance in the ABL is

$$ \left( \frac{\text{Pr}_t e^\xi}{\kappa v_\tau^*} \right) \frac{\partial (\rho^* T^*)}{\partial t^*} = \frac{\partial}{\partial \xi} \left( \frac{\rho^*}{\phi_h} \frac{\partial T^*}{\partial \xi} \right), \qquad (\text{D3}) $$

where we define $\xi \equiv \ln z$. Because $\rho^*T^* = p^*$ is invariant at constant atmospheric pressure $p^* \equiv p/p_{\text{st}}$, Eq. (D3) has no unsteady term, and the flux is independent of $z$. Integrating it between the surface at $z = z_0$ (index $s$) and the altitude $z_T$ where ambient temperature $T_T$ is measured, we find

$$ T^*(z^*) = T_s^* \exp \left[ \frac{\Phi_{2h}(z^*/L^*)}{\Phi_{2h}(z_T^*/L^*)} \ln \left( \frac{T_T^*}{T_s^*} \right) \right], \qquad (\text{D4}) $$
1272 where we take $T_s$ to be the temperature measured at the buried sensor closest to the surface, and the integral function is

1273
$$ \Phi_{2h}(z) = \int_{\xi=z_0}^{z} \phi_h d \ln \xi. \qquad (D5) $$

1275 From Eq. (D1), the dimensionless heat flux is then

1276
$$ q^* = \frac{\kappa v_T^* p^*}{\mathrm{Pr}_t \Phi_{2h} (z_T^*/L^*)} \ln \left( \frac{T_T^*}{T_s^*} \right). \qquad (D6) $$

1277 Meanwhile, the Monin-Obukhov length in Eq. (55), dimensionless with $(\alpha J)^{1/2}$, satisfies the equation

1279
$$ L^* = \left( \frac{\alpha}{J^3 g^2} \right)^{1/2} \frac{v_T^{*2} \mathrm{Pr}_t \Phi_{2h} (z_T^*/L^*)}{\kappa^2 \ln(T_T^*/T_s^*)}, \qquad (D7) $$

1280 which we solve for $L^*$ using time-histories of the measured wind speed $U$, temperatures $T_T$ and $T_s$ reported in section 4, $v_T/U \simeq 0.036$ measured at the probe location in January 2017, and adopting $\mathrm{Pr}_t \simeq 1$.

## Appendix E Soret coefficient

1284 Chapman and Cowling (1953) used the kinetic theory to derive the Soret diffusion mass flux $-\rho(D/\bar{\omega})k_T \nabla \ln T$ of a gas driven by a temperature gradient. They found

1286
$$ k_T = 5(C_s - 1) \frac{S_1[n_1/(n_1 + n_2)] - S_2[n_2/(n_1 + n_2)]}{Q_1(n_1/n_2) + Q_2(n_2/n_1) + Q_{12}}, \qquad (E1) $$

1287 where $n_1 m_1$ is the partial density of gas 1 with molecular mass $m_1$ diffusing into gas 2. Substituting 1 for 2 and vice-versa completes the set of equations. With water vapor (gas 1) dilute in air (gas 2), $n_1 m_1 = \rho Y$ and $n_2 m_2 = (1 - Y)\rho \simeq \rho$. Using the shorthand $M_1 \equiv m_1/(m_1 + m_2)$, Chapman and Cowling (1953) calculated

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$$ S_2 = M_2^2 E_2 - 3M_1(M_1 - M_2) - 4M_2 M_1 A_s \qquad (E2) $$

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$$ Q_2 = M_2 E_2 [6M_1^2 + (5 - 4B_s) M_2^2 + 8M_2 M_1 A_s] $$

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$$ Q_{12} = 3(M_1 - M_2)^2(5 - 4B_s) + 4M_1 M_2 A_s (11 - 4B_s) + 2M_1 M_2 E_1 E_2 $$

1294
$$ E_2 = \frac{2\sqrt{2}}{5M_2M_1^{1/2}} \left(\frac{\sigma_2}{\sigma_{12}}\right)^2 $$

1295
$$ \sigma_{12} = (\sigma_1 + \sigma_2)/2, $$

1296 with $A_s = 2/5$, $B_s = 3/5$ and $C_s = 6/5$. Because $n_1/n_2 \simeq M^*Y \ll 1$, where $M^*$ is the ratio of the molar masses of air and water, Eq. (E1) simplifies to $k_T \simeq -S_2 M^* Y / Q_2 \equiv a_T Y$. With $M_1 = 1/(1 + M^*)$, $M_2/M_1 = M^*$, $\sigma_1 \simeq 2.65 \AA$, $\sigma_2 \simeq 3.60 \AA$, we find $a_T \simeq -0.319$.

1300 With this magnitude, our calculations indicate that log derivatives of temperature $\partial \ln T^*/\partial x^*$ and $\partial \ln T^*/\partial t^*$ are too small for terms involving $a_T$ to play a significant role in Eq. (49). Therefore in our case, the Soret effect can be ignored. In extra-terrestrial applications with stronger temperature variations, this may no longer be the case.

## Appendix F Seepage profiles

1305 On time scales long enough to establish equilibrium, the mass conservation Eq. (15) and the rate Eq. (24) can be manipulated to extract an ODE for the evolution of $u^*$ along
the relative depth $x^*$,

$$ \frac{\partial u^*}{\partial x^*} = \left[ -\frac{\partial \ln p^*}{\partial t^*} + \frac{\partial \ln T^*}{\partial t^*} + \frac{(M^* - 1)}{1 + Y(M^* - 1)} \frac{\partial Y}{\partial t^*} \right] \quad (F1) $$
$$ \begin{aligned} & - u^* \left[ \frac{\partial \ln p^*}{\partial x^*} - \frac{\partial \ln T^*}{\partial x^*} - \frac{(M^* - 1)}{1 + Y(M^* - 1)} \frac{\partial Y}{\partial x^*} \right] \\ & - \mathbb{R}\Omega_1 \frac{T^*}{p^*} \left[ f_T \frac{\partial T^*}{\partial t^*} + f_Y \frac{\partial Y}{\partial t^*} \right], \end{aligned} $$

which is coupled with Darcy's law (17) through the dimensionless term

$$ \frac{\partial \ln p^*}{\partial x^*} = - \left( \frac{\mu \alpha}{K p_{st}} \right) \frac{u^*}{p^*}. \quad (F2) $$

These equations are solved with ODE (30), or with PDE (26). They are used to calculate profiles of $u^*$ in the insets of Fig. 7.

## Appendix G Kinetic limit

In this Appendix, we estimate the time $\tau_d$ needed to bring the mass fraction $\Omega_d$ held by a spherical grain of diameter $d$ to its value $\Omega_{ed}$ at equilibrium with the imposed $Y$ around it. This equilibration process is driven by the volumetric rate of evaporation at the particle scale, which is governed by the surface mass flux $\dot{W}_s''$ expanding into the interstitial space of density $\rho(1-\nu)$ surrounding grains. In Eq. (44), $Y_e$ is the mass fraction at equilibrium with the instantaneous $\Omega_d$ on the solid of size class $d$ given by Eq. (B1).

As in the combustion of a solid particle (Kanury, 1975), there are two limiting regimes. In the “kinetic limit”, $k_m$ is small relative to $\rho D/d$, so diffusion erases gradients of $Y$, and the flux is given by Eq. (43), $\dot{W}_s'' = \dot{W}_{HKS}''$. Combining with Eq. (44), the characteristic time of size class $d$ is

$$ \tau_d = \tau_{HKS} = \frac{\rho(1-\nu)d}{6\nu k_m} = \left(\frac{1-\nu}{\nu}\right) d \sqrt{\frac{\pi M W_{H_2O}}{18RT}} \left(\frac{1-\kappa_c/2}{\kappa_e}\right), \quad (G1) $$

which, upon averaging over the particle size distribution, would return the same expression with the Sauter diameter $d_{2,3}$ instead of $d$.

If instead diffusion is not as fast, it is essential to involve it in the local water mass balance, as the local vapor mass fraction $\tilde{Y}$ may vary within each pore. For simplicity, we imagine, crudely, that particles are isolated, as commonly assumed to derive closed-form solutions for solid or droplet combustion (Kanury, 1975). Under quasi-steady conditions $\partial\tilde{Y}/\partial t \simeq 0$, the mass flow rate through any spherical shell at radius $r$ from the particle center is invariant and equal to its value $\pi d^2 \dot{W}_s''$ at the surface. Then, the isothermal balance of water vapor mass in a slice $[r, r+dr]$ is

$$ \pi d^2 \dot{W}_s'' \frac{\partial \tilde{Y}}{\partial r} - \rho D \frac{\partial}{\partial r} \left( r^2 \frac{\partial \tilde{Y}}{\partial r} \right) = 0. \quad (G2) $$

At $r = d/2$, the net evaporation rate of a unit area of water film must balance the sum of the advection and diffusion fluxes in the gas phase,

$$ \dot{W}_s'' = \dot{W}_s'' Y_s - \rho D \left( \frac{\partial \tilde{Y}}{\partial r} \right)_s. \quad (G3) $$

Integrating Eq. (G2) twice subject to $\tilde{Y} \to Y$ at $r \to \infty$ and boundary condition (G3), we find

$$ (1 - \tilde{Y}) = (1 - Y) \exp[-d^2 \dot{W}_s''/(4\rho Dr)], \quad (G4) $$
where $\dot{W}_s''$ is given by Eq. (43). Applying this profile expression to the surface at $r = d/2$ and eliminating $Y_s$ using Eq. (43) yields an equation to solve for $\dot{W}_s''$ in the general case,

$$1 - Y_e + \frac{\dot{W}_s''}{k_m} = (1 - Y) \exp \left( - \frac{d \dot{W}_s''}{2 \rho D} \right). \qquad (G5)$$

As expected, we recover $\dot{W}_s'' = \dot{W}_{HKS}''$ in the kinetic limit by making $D \to \infty$. In the “diffusion limit”, slow diffusion is the rate limiting step. Here, by making $k_m \to \infty$, $\dot{W}_s''$ becomes

$$\dot{W}_{\text{DIFF}}'' = \frac{2\rho D}{d} \ln \left( \frac{1-Y}{1-Y_e} \right) \approx \frac{2\rho D}{d} (Y_e - Y), \qquad (G6)$$

with characteristic time

$$\tau_{\text{DIFF}} = (1 - \nu)d^2/(12\nu D). \qquad (G7)$$

Because an interstitial space with $\nu > 0.5$ has a pore size on the order of $d[(1-\nu)/\nu]^{1/3} < d$, the time to diffuse vapor through this space should be even less than predicted by Eq. (G7). Combining the profile Eq. (G4) with Eq. (G6), we also find $Y_s = Y_e$ in the diffusion limit.

Expanding the exponential to first order in Eq. (G5) and recognizing that $Y \ll 1$, the flux in the general case is approximately given by

$$\frac{1}{\dot{W}_s''} \approx \frac{1}{\dot{W}_{\text{DIFF}}''} + \frac{1}{\dot{W}_{\text{HKS}}''}, \qquad (G8)$$

confirming that the two limiting processes operate in parallel. Damköhler’s “second ratio” $Da_{II} \equiv \tau_{\text{DIFF}}/\tau_{\text{HKS}} = k_m d/(2\rho D)$ reveals which mechanism predominates by being the slowest. A large $Da_{II}$ implies the diffusion limit. With a unity accommodation coefficient of evaporation ($\kappa_e \sim 1$), this would likely be the case, since $Da_{II} \sim 860$ for our mean Sauter diameter under standard conditions. For example, to complement the capillary condensation model of Philip (1964), Shahraeeni and Or (2010) postulated a diffusion-limited process for film adsorption at the microscopic scale of wedge-shaped grain asperities. However, as section 12 shows, the characteristic time involved $\tau_{\text{DIFF}} \sim 240 \mu s$ would be much too small to explain the exponential decay of $\Delta Y$ with depth. Therefore, the behavior of subsurface waves imply instead that the kinetic limit prevails at the grain scale.

## Appendix H Developing mass transfer boundary layer

We invoke integral equations of the boundary layer to model wind-driven convective mass transfer, coupled to the advection/diffusion measured through the sand surface, with possible mass flux exchanged with the ABL aloft. Unlike Appendix D, where we entertained unsteady terms, here we ignore them due to the more rapid development of the turbulent mass transfer boundary layer (TBL). Overall mass conservation in a TBL slice perpendicular to the surface is

$$\frac{d}{d\mathcal{X}} \int_{z=0}^{z_{\ell}} \rho U dz - \rho_{\ell} v_{\ell} + \rho_s v_s = 0, \qquad (H1)$$

where $\mathcal{X}$ is distance along the wind from the leading toe of the dune, $v$ is superficial gas velocity ($> 0$ toward sand), and indices $s$ and $\ell$ represent the surface and the top of the slice, respectively. The corresponding mass balance of water vapor is

$$\frac{d}{d\mathcal{X}} \int_{z=0}^{z_{\ell}} \rho U Y dz - \rho_{\ell} Y_{\ell} v_{\ell} + \rho_s Y_s v_s - \dot{\Psi}_{\ell}'' + \dot{\Psi}_s'' = 0, \qquad (H2)$$

where vapor fluxes $\dot{\Psi}''$ are $> 0$ into sand. Eliminating $\rho_{\ell}v_{\ell}$ from Eq. (H2) using Eq. (H1), dividing by $\rho_{\ell}U_{\ell}$ and letting $\ell \to \infty$, we find

$$\frac{d\delta_m}{d\mathcal{X}} + \frac{\rho_s v_s}{\rho_\infty U_\infty} - \frac{\dot{\Psi}_\infty''}{\rho_\infty U_\infty (Y_s - Y_\infty)} + \frac{\dot{\Psi}_s''}{\rho_\infty U_\infty (Y_s - Y_\infty)} = 0, \qquad (H3)$$
1385 where $\rho_\infty$ and $U_\infty$ are recorded at an altitude higher than the mass transfer thickness of the boundary layer

1386
$\delta_m \equiv \int_{z=0}^{\infty} \frac{\rho U}{\rho_{\infty} U_{\infty}} \left( \frac{Y - Y_{\infty}}{Y_s - Y_{\infty}} \right) dz.$
(H4)

1388 Because water vapor has a diffusion coefficient $D$ on the same order as the thermal diffusivity of air or, equivalently, that its Lewis number is near unity, an analogy with heat transfer in the TBL suggests that $d\delta_m/d\chi$ is a kind of Stanton number

1391
$$ \text{St}_X = \frac{d\delta_m}{d\chi} \approx \frac{a_m}{\text{Re}_X^{1/5} \text{Sc}^{0.4}}, \qquad (\text{H5}) $$

1392 where $\text{Sc} \equiv (\mu/\rho)/D \approx 0.60$ is the Schmidt number of water vapor diffusing in air.
1393 Then, from its standpoint, the ABL receives a net surface flux

1394
$$ \dot{\Sigma}'' \equiv -\dot{\Psi}_s'' - \rho_s v_s (Y_s - Y_a) = a_m \frac{\rho_\infty U (Y_s - Y_a)}{\text{Re}_X^{1/5} \text{Sc}^{0.4}} + \dot{\Sigma}_\infty'', \qquad (\text{H6}) $$

1395 where $\dot{\Sigma}''_\infty \equiv -\dot{\Psi}''_\infty$ is the upward moisture flux on top of the TBL, with both $\dot{\Sigma}''$ and $\dot{\Sigma}''_\infty$ now counted > 0 upwards, and we identify $Y_\infty$ and $U_\infty$ as the mass fraction $Y_a$ and wind speed $U$ recorded by the top instruments of the weather station. Using this equation and our net flux measurements provided in dimensionless form in Eqs. (53)-(32), we estimate the coefficient $a_m$ and the exchange flux $\dot{\Sigma}''_\infty$ toward the ABL aloft (section 13).

1400 Acknowledgments

We are grateful to Patrick Perré, Floran Pierre, and Joël Casalinho for help measuring sand isotherms; to Jean-Luc Métayer for measuring particle size distribution, to Anthony Hay, Ali Sultan, Renée Richer, Christopher Ogden, Dah Ould Ahmedou, Pascal Dupont, Anushree Acharya, Sara Abdul-Majid, Osama Al Achek, Carlos Mejia, Nathalie Vriend, and Matthew Arran for assistance in the field; to Chun-Chieh (Jimmy) Chang for creating special batteries; to Robert Foster and Hai Trinh for advice on capacitance instrumentation; to Stephen Keast for help with probe design; to Marc Massot, Greg Bewley, Abraham Stroock, Jean-Yves Parlange, Tammo Steenhuis, Shmuel Assouline, Dani Or, Michael Gentzler and Sylvain Dupont for illuminating discussions; to Isabelle Louge and J. Gregory Butler for vetting the plain language summary; and to Amy East, Thomas Pähtz, Giora Kidron and anonymous reviewers for their insight on revising the manuscript. This paper was made possible by the support of NPRP grants 09-546-2-206 and 6-059-2-023 from the Qatar National Research Fund, and by a Qatar Foundation Research Excellence Award.

The entire data set, movie animations, original figures and nomenclature are available as supporting information at doi:10.7298/kqgg-5888. The “ReadMeDataFilesContents” file lists their contents and specifies all parameters adopted in this work. The movies animating Figs. 4, 6 and 8 are listed in the “ReadMeSupportingMovie” file.

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