Evaporation-Driven Density Instabilities in Saturated Porous Media

Abstract

Soil salinization is a major cause of soil degradation and hampers plant growth. For soils saturated with saline water, the evaporation of water induces accumulation of salt near the top of the soil. The remaining liquid gets an increasingly larger density due to the accumulation of salt, giving a gravitationally unstable situation, where instabilities in the form of fingers can form. These fingers can, hence, lead to a net downward transport of salt. We here investigate the appearance of these fingers through a linear stability analysis and through numerical simulations. The linear stability analysis gives criteria for onset of instabilities for a large range of parameters. Simulations using a set of parameters give information also about the development of the fingers after onset. With this knowledge, we can predict whether and when the instabilities occur, and their effect on the salt concentration development near the top boundary.

1 Introduction

Evaporation of saline water from soils can cause accumulation of salts in the upper part of the soil, which has a large environmental impact as it hampers plant growth and affects biological activities (Daliakopoulos et al. 2016). As water evaporates, the salts accumulate near the top of the porous medium, which has a negative impact on root water uptake (Chaves et al. 2008). If the solubility limit of the salt is exceeded, the salts precipitate. In this case, a salt crust at the top of the soil is formed, disconnecting the soil from the atmosphere (Chen 1992; Jambhekar et al. 2015; Mejri et al. 2017). The appearance of the salt crust strongly affects the growth conditions for many agricultural plants (Pitman and Läuchli 2002; Singh 2016). In arid regions, for example in Tunisia, soil salinization and soil crust formation are already interfering with and disabling agricultural activities (Mejri et al. 2020). This is not a new problem, but is gradually causing a greater impact as larger areas are affected (Vereecken et al. 2009).

Different observations in natural systems show that evaporation processes from soils cause an accumulation of salts in the upper part of the soil (Allison and Barnes 1985). This can again lead to salt precipitation, creating salt lakes and/or density-driven currents due to varying salt concentrations (Duffy and Al-Hassan 1988; Geng and Boufadel 2017). As salts accumulate at the top of the soil due to water evaporating, the density of the remaining liquid increases with increased salt concentrations (Geng and Boufadel 2015, 2017). This may lead to a gravitationally unstable setting since the liquid near the top of the soil is the heaviest, due to the accumulated salts (Gilman and Bear 1996; Nield and Bejan 2017; Wooding et al. 1997). When the soil is permeable, density instabilities in the form of fingers can be triggered (Wooding et al. 1997), which is also relevant in the context of CO$_2$ storage (Elenius et al. 2012; Riaz et al. 2006). The formation of density instabilities in the form of fingers induces a downward transport of the accumulated salts from the upper part of the soil toward the lower part, where the salt concentration is lower. Hence, when the density instabilities develop, they can hinder the salt concentrations near the top of the soil to exceed its solubility limit. However, for soils of low permeability, these instabilities will typically not develop, or they develop at a later time. Under these conditions, salts will continue to accumulate until salt precipitates, and a salt crust at the top of the soil is formed. This means that the occurrence of salt precipitation is tightly connected to the development of convective instabilities. Hence, understanding the process of soil salinization and the interplay with density instabilities is key questions to prevent degradation of soil quality and to ensure food production (Shokri et al. 2010; Shokri-Kuehni et al. 2020).

It is well-known that the question of whether the density instabilities occur, can be addressed by a linear stability analysis. Such an analysis has been applied to a wide range of porous-media problems where the density difference creates a gravitationally unstable setting (Nield and Bejan 2017). The onset of instabilities where an increased salt concentration at the top of the porous domain triggers the instabilities, is analyzed in Elenius et al. (2012); Riaz et al. (2006); van Duijn et al. (2019). Fingers are found to appear when the strength of the density difference overcomes the resistance of the porous medium. This is usually expressed and quantified through a critical threshold of the Rayleigh number, such that when the Rayleigh number is larger than this threshold, instabilities can occur. A density difference, and hence a change in salt concentration, is needed to induce instabilities. Hence, a strong diffusion would hinder the density difference to be strong enough, as the concentration profile is smoothed. A large resistance of the porous medium, which corresponds to a small permeability, makes it more difficult for the density difference to trigger instabilities.

The above-mentioned works on salt-induced instabilities consider a prescribed salt concentration or a prescribed density on the top boundary (Elenius et al. 2012; Riaz et al. 2006; van Duijn et al. 2019). When considering evaporation from a porous medium, the salt concentration at the top boundary develops with time as the water gradually evaporates and the dissolved salts remain. As we will see later, this can be modeled with a Robin-type boundary condition for the salt, which means that the value of the salt concentration is connected to its gradient at the top boundary. Such boundary conditions have been considered in other linear stability problems, see e.g., Barletta et al. (2009); Hattori et al. (2015).

In this work, we consider soils that remain fully saturated with water throughout the evaporation. This means that we consider a case where the modeled porous medium is connected to a deeper groundwater aquifer, which is also fully saturated. The deeper groundwater aquifer supplies the modeled soil with water during the evaporation. We also assume that the capillary pressure of the system remains below the entry pressure of the soil, such that the soil remains fully saturated. If conditions of partial saturation occur, it would be necessary to include also the flow of air through the unsaturated zone, or to use Richards equation. Evaporation would typically lead to an unsaturated zone in the upper part of the porous medium, which again has an impact on the evolution of the salt concentration (Shokri et al. 2010; Shokri-Kuehni et al. 2020).

Following the above assumptions, the evaporation of water induces a vertical, upward throughflow through the domain. The effect of upward throughflow with a given density difference between the top and bottom boundaries has been found to have a stabilizing effect on the onset of instabilities (Homsy and Sherwood 1976; van Duijn et al. 2002). That means, a stronger upward throughflow would increase the critical Rayleigh number, making it more difficult for the density difference to trigger the formation of downward-flowing fingers. It is, hence, not obvious whether an increased evaporation flux of water would have a stabilizing or destabilizing effect: an increased evaporation leads to an increased upward throughflow, which stabilizes the system, but at the same time the accumulation of salts near the top boundary is increased, which destabilizes.

The method of linear stability gives estimates for the onset of gravitational instabilities and for the time of their appearance, by considering a simplified system of equations. However, estimates for a large range of parameters can be found at low costs. After the instabilities have formed, one has to rely on numerical simulations of the governing model equations to address the further development of the salt concentration. Numerical simulations can give information on the strength and shape of the appearing convection pattern, as well as their effect on the salt transport and salt precipitation. However, these numerical simulations are expensive and need to be performed on bounded domains. Although we consider a simplified setup by assuming fully saturated conditions, this proposed analysis gives valuable insight for this idealized case and creates a starting point for further analysis when incorporating an unsaturated zone in the future.

This paper is organized as follows. In Sect. 2, we formulate the general model equations together with initial and boundary condition to address evaporation from a porous medium saturated with saline water. In Sect. 3, we consider a simplified model, for which we perform a linear stability analysis, giving criteria for when instabilities can occur. Section 4 explains the numerical framework used to simulate the general model. The results from the linear stability analysis and the numerical simulations are compared and discussed in Sect. 5, before final remarks are given in Sect. 6.

Table 1 Nomenclature

2 Mathematical Model

This section describes the physical assumptions, the domain, the partial differential equations and the boundary and initial conditions which form the mathematical model considered in this paper to describe evaporation from the top of a porous medium and the subsequent changes within the medium. Figure 1 sketches the domain together with the most important model choices. All variables and parameters are summarized in Table 1.

Fig. 1

Sketch of evaporation from porous medium and effect on salt concentration

2.1 Domain and Model Equations

The considered domain is unbounded in the vertical direction and either bounded or unbounded in the horizontal directions. Specifically, we consider either

$$\begin{aligned} \varOmega = \{ (x,y,z)\in {\mathbb {R}} : z>0\}. \end{aligned}$$

(1)

or

$$\begin{aligned} \varOmega _{{\mathcal {W}}} = \{ (x,y,z)\in {\mathbb {R}} : |x|,|y|<{\mathcal {W}}, z>0\}, \end{aligned}$$

(2)

where ${\mathcal {W}}$ denotes the horizontal half width. Note that the positive vertical direction is pointing downward: hence, $z=0$ indicates the top of the domain. Within our domain, we consider mass conservation of water and salt, along with Darcy’s law representing the momentum conservation. The porous medium is assumed to be fully saturated with liquid, and the liquid consists of water and dissolved salt. Exemplary, the salt sodium chloride NaCl is used. We formulate the conservation of each of these two components independently. Both water and salt are advected with the liquid’s velocity, and both are subject to diffusion:

$$\begin{aligned} \partial _t (\phi \rho _\mathrm {mol} {\mathsf {x}}^\kappa ) = \nabla \cdot \left( - \rho _\mathrm {mol} {\mathsf {x}}^\kappa {\mathbf {Q}} + D \rho _\mathrm {mol}M \nabla \big ( \frac{{\mathsf {x}}^\kappa }{M}\big ) \right) + r^\kappa . \end{aligned}$$

(3)

Here, $\kappa \in \{\mathrm {w, NaCl} \}$ represents the two components of the liquid phase, and ${\mathsf {x}}^\kappa$ denotes the mole fraction of the component $\kappa$. Hence, the sum of these two mole fractions is by definition 1, which is also reflected in (3). Further, $\phi$ is porosity, $\rho _\mathrm {mol}$ is the molar density of the liquid phase, and M is the molar mass of the liquid mixture. Finally, D is the effective diffusivity of the components in the mixture. Note that D represents the diffusion of water mixed with salt, which is why we use the same diffusion coefficient for both components. A large diffusivity D would lead to disturbances and gradients in the concentration fields being quickly smoothed away. The reaction term $r^\kappa$ accounts for chemical reactions inside the domain and is only non-zero when salt precipitation takes place within the porous medium. This means, $r^\mathrm {w} =0$ while

$$\begin{aligned} r^\mathrm {NaCl} = {\left\{ \begin{array}{ll} 0 &{} \text { when } {\mathsf {x}}^\mathrm {NaCl}\le {\mathsf {x}}^\mathrm {NaCl}_{\text {max}}, \\ <0 &{} \text { when } {\mathsf {x}}^\mathrm {NaCl}>{\mathsf {x}}^\mathrm {NaCl}_{\text {max}},\end{array}\right. } \end{aligned}$$

(4)

where ${\mathsf {x}}^\mathrm {NaCl}_{\text {max}}$ is the solubility limit of NaCl. Hence, we have a sink term for NaCl in (3) when ${\mathsf {x}}^\mathrm {NaCl}$ exceeds its solubility limit. The Darcy flux ${\mathbf {Q}}$ is given by

$$\begin{aligned} {\mathbf {Q}} = -\frac{K}{\mu } (\nabla P - \rho g {\mathbf {e}}_z ), \end{aligned}$$

(5)

where the permeability K is assumed to be a scalar, as the porous medium is assumed to be isotropic. The liquid viscosity $\mu$ is assumed to be constant. Here we use the mass density $\rho$ of the liquid phase. Finally, P is the pressure, and g is gravity. Note that the unit vector ${\mathbf {e}}_z$ points downward. Also note that a larger density of the liquid supports a stronger downward flow.

We assume that the liquid density varies with the salt mass fraction $X^\mathrm {NaCl}$ through the linear dependence

$$\begin{aligned} \rho (X^{\mathrm {NaCl}}) = \rho _{0}(1+\gamma (X^{\mathrm {NaCl}}-X^{\mathrm {NaCl}}_0)), \end{aligned}$$

(6)

where $\rho _{0}$ and $X^{\mathrm {NaCl}}_0$ are the initial liquid density and salt mass fraction, and $\gamma$ is a volumetric constant. Note that the initial salt mass fraction is a constant and hence corresponds to a uniform salt distribution in space. The conversion between molar density and mass density is through

$$\begin{aligned} \rho =\rho _{\text {mol}}\cdot M, \end{aligned}$$

(7)

where the molar mass M of the liquid mixture also depends on the salt content. Both molar and mass liquid density increase with the salt concentration. Note that if a different type of solute is considered, expression (6) still applies, although with a different value of $\gamma$.

If the salt exceeds its solubility limit, salt precipitates and becomes part of the solid. In this case, the porosity $\phi$ will change with time, and we apply a mole balance for the solid salt to describe this process:

$$\begin{aligned} \rho _\mathrm {mol,solid}~ \partial _t \phi = - r_\mathrm {solid}. \end{aligned}$$

(8)

Here, $\rho _\mathrm {mol,solid}$ is the molar density of the solid salt phase and $r_\mathrm {solid}$ the reaction term. We have that $r_\mathrm {solid}=-r^{\mathrm {NaCl}}$. When there is precipitation, the porosity of the porous medium and thus the permeability decreases. For the calculation of the permeability, a Kozeny-Carman-type relationship based on the initial values $\phi _0$, $K_0$, is used:

$$\begin{aligned} K = K_0\left( \frac{1-\phi _0}{1-\phi } \right) ^2 \left( \frac{\phi }{\phi _0} \right) ^3. \end{aligned}$$

(9)

Note that as long the salt remains below the solubility limit, there is no precipitation, and $\phi$, K remain equal to their initial values $\phi _0$, $K_0$. These initial values are constants, and hence correspond to an initially homogeneous porous medium.

2.2 Discussion of Physical Processes

Fig. 2

Relevant forces and fluxes for the development of instabilities

In Fig. 2, the relevant fluxes described by Eq. (3) and (5) are illustrated in the context of the development of instabilities. Darcy’s law describes the convective flux depending on a pressure gradient force and a gravitational force. In the considered setup, the pressure gradient generates an upward force due to evaporation at the top, while the gravitational force, however, points downward. Hence, the resulting convective flux depends on the balance of the two counter-effective forces.

Considering a perturbation with increased salt concentration and liquid density, the increased density induces stronger gravitational forces. In case of a still dominating pressure gradient force, the convective flux is in upward direction but slowed down at the location of the perturbation. This leads to a compensation of the flux by the fluxes from the surrounding, which accumulates salt and thus enhances the perturbation. This means that the salt accumulation increases the density and gravitational force even more, which can lead to a dominating gravitational force. In this case, a resulting convective downward flux is generated, which leads to a development of so-called fingers which transport the accumulated salt downward. In addition, diffusive fluxes are considered in Eq. (3). The development of instabilities is counteracted by the diffusive transport, which tries to balance out the concentration differences. Hence, the balance of the counteracting convective and diffusive fluxes determines if and how fast instabilities develop.

Parameters like the permeability for the porous medium, the diffusion coefficient for the fluid mixture or the evaporation rate as boundary condition have also an important influence on the development of the instabilities. In case of higher permeabilities, for example the convective flux is enhanced compared to the diffusive flux. This leads to a faster development of the instabilities.

2.3 Initial and Boundary Conditions

As initial conditions, we take, see (6) and (8),

$$\begin{aligned} {\mathsf {x}}^\mathrm {NaCl}|_{t=0}&= {\mathsf {x}}_0^\mathrm {NaCl}, \end{aligned}$$

(10)

$$\begin{aligned} \phi |_{t=0}&= \phi _0. \end{aligned}$$

(11)

At the top of the domain, we allow water to evaporate while salt remains behind. This corresponds to specifying a given molar flux $E_\mathrm {mol}$ for the water component, while a zero flux is considered for NaCl:

$$\begin{aligned} \left( \rho _\mathrm {mol} {\mathsf {x}}^\mathrm {w} {\mathbf {Q}} - D \rho _\mathrm {mol} \nabla {\mathsf {x}}^\mathrm {w} \right) \cdot {\mathbf {e}}_z|_{z=0}&= -E_\mathrm {mol},\end{aligned}$$

(12)

$$\begin{aligned} \left( \rho _\mathrm {mol} {\mathsf {x}}^\mathrm {NaCl} {\mathbf {Q}} - D \rho _\mathrm {mol} \nabla {\mathsf {x}}^\mathrm {NaCl} \right) \cdot {\mathbf {e}}_z|_{z=0}&= 0. \end{aligned}$$

(13)

Since the vertical unit vector ${\mathbf {e}}_z$ points downward, the evaporation flux $E_\mathrm {mol}>0$ corresponds to an upward flux of water. In general, the Darcy flux ${\mathbf {Q}}$ will be non-zero on the top boundary due to the presence of the evaporative flux. This also means that the no-flux boundary for salt (13) is a Robin boundary condition. Since water escapes through the top boundary while salt remains behind, we expect an accumulation of salt concentration near the top boundary. Increasing $E_\mathrm {mol}>0$ yields the concentration of NaCl increasing faster near the top boundary. We will here assume a prescribed evaporation flux that is constant with respect to both time and space. This is a simplification, especially when time periods of more than a few hours are considered (Heck et al 2020) and should in this case be interpreted as an averaged evaporation flux.

Since the domain is semi-infinite in the vertical direction, boundary conditions at the bottom boundary are not needed. There the values are inherited from the initial conditions.

For the horizontal extent of the domain, we separate between the bounded and unbounded case. In the unbounded case, we do not need any boundary conditions in the horizontal direction. For the bounded case, we apply no-flux boundary conditions at the vertical walls. This corresponds to

$$\begin{aligned} {\mathbf {Q}}\cdot {\mathbf {n}}|_{x,y=\pm {\mathcal {W}}} = 0,\quad \nabla {\mathsf {x}}^\kappa \cdot {\mathbf {n}}|_{x,y=\pm {\mathcal {W}}} = 0, \end{aligned}$$

(14)

where ${\mathbf {n}}$ is the horizontal unit vector pointing out of the sidewalls of the bounded domain $\varOmega _{{\mathcal {W}}}$.

The general equations formulated here form the starting point for our further investigation of evaporation from the porous medium and subsequent onset of density instabilities. For the linear stability analysis in Sect. 3, we use a slightly simplified system of equations and cast the equations in non-dimensional form to keep the analysis as general as possible. The numerical setup described in Sect. 4 uses the model as described above, but using bounded domains. The comparison between the two approaches in Sect. 5 considers the dimensional case in order to connect back to the physical problem.

3 Linear Stability Analysis

To address when instabilities can occur and give a criterion for the onset of instabilities depending on the model parameters, we perform a linear stability analysis following relatively standard steps, but giving a non-standard outcome due to the special setup following the evaporation from the top boundary.

To accommodate the linear stability analysis, we consider here slightly simplified model equations than the general presented in Sect. 2. These simplified equations are non-dimensionalized to find ratio of parameters that characterize the overall behavior. The equations are non-dimensionalized using identified reference values characterizing the setup. As we have somewhat untypical boundary conditions due to the evaporation at the top boundary, the chosen reference values are non-standard. From the non-dimensional formulation, we derive a time-dependent stable solution ("the ground state"). Due to the gradually growing salt concentration following the evaporation, the salt ground state is unbounded and does not exhibit a steady state. For finite times, the time-dependent ground state is perturbed using the frozen-profile approach, in order to address its stability. Since the perturbed quantities are small, we linearize the equations for the perturbed quantities. The linearized perturbation equations are then finally formulated as an eigenvalue problem, which is solved numerically. This will give information on the stability of the ground state as a function of time.

Note that the linear stability analysis is made general in the sense that it is also applicable for salt types other than NaCl by adjusting the value of corresponding parameters, in particular of the volumetric constant $\gamma$.

3.1 Simplified Model Equations

We apply the method of linear stability for a simplified version of the model presented in Sect. 2. We assume that the salt is completely dissolved and that its mass fraction is small compared to the mass fraction of water. We take advantage of the fact that ${\mathsf {x}}^\mathrm {w}\approx 1$ and hence simplify the mass conservation equation for water. Then we invoke the Boussinesq approximation, meaning that the liquid density $\rho$ can be considered constant except in the gravity term of Darcy’s law. Further, we disregard salt precipitation, which means that the simplified model is only valid up to the salt reaches its solubility limit. Porosity and permeability will, hence, remain constant. With these simplifications, Eq. (3) reduce to

$$\begin{aligned} \nabla \cdot {\mathbf {Q}}&= 0,\end{aligned}$$

(15)

$$\begin{aligned} \phi \partial _tX&= \nabla \cdot (-{\mathbf {Q}}{\mathbf {X}}+D\nabla X), \end{aligned}$$

(16)

where X is the mass fraction of salt. Since only salt mass fraction is used as a variable in the following, we skip the superscript. Darcy’s law is kept as in (5), and the density of the liquid is still depending linearly on the salt concentration (6). Note that these model equations are simplified since varying water mass fraction is not accounted for, and since we apply the Boussinesq approximation. However, for relatively low variability in the salt content (i.e., below the solubility limit of salt), these assumptions are reasonable.

As initial condition for the salt, we use the corresponding version of (10), namely

$$\begin{aligned} X|_{t=0}&= X_0, \end{aligned}$$

(17)

where $X_0$ is the initial mass fraction, and the boundary conditions (14) at the sidewalls are, hence, correspondingly formulated for X.

Due to the difference in addressing the mass of water and salt, the boundary conditions (12) and (13) are replaced with

$$\begin{aligned} {\mathbf {Q}}|_{z=0}&= -E{\mathbf {e}}_z \end{aligned}$$

(18)

$$\begin{aligned} (X{\mathbf {Q}}-D\nabla X)|_{z=0}\cdot {\mathbf {e}}_z&= 0, \end{aligned}$$

(19)

where E is the evaporation rate in terms of a volume flux of water. The two evaporation fluxes $E_{\mathrm {mol}}$ and E are related through

$$\begin{aligned} E_{\mathrm {mol}} = E\cdot M\cdot \rho . \end{aligned}$$

(20)

For convenience and to identify parameter dependencies, we will recast the equations in a dimensionless form.

3.2 Non-dimensional Model

The corresponding non-dimensional variables are denoted by a hat:

$$\begin{aligned} ({\hat{x}},{\hat{y}},{\hat{z}}) =&\frac{(x,y,z)}{\ell _{\text {ref}}},\quad {\hat{t}} = \frac{t}{t_{\text {ref}}}\quad {\hat{P}} = \frac{P}{P_{\text {ref}}},\end{aligned}$$

(21)

$$\begin{aligned} \hat{{\mathbf {Q}}} =&\frac{{\mathbf {Q}}}{Q_{\text {ref}}},\quad {\hat{\rho }} = \frac{\rho }{\rho _\text {ref}} \quad {\hat{X}} = \frac{X}{X_{\text {ref}}}. \end{aligned}$$

(22)

We choose reference quantities that are meaningful to address the effect of evaporation, taking into account that the domain is vertically unbounded and that no prescribed density difference is given.

As length reference, we choose an intrinsic length, namely the ratio between diffusion and evaporation $\ell _{\text {ref}} = D/E$, which quantifies the length scale at which diffusion can smooth out concentration differences caused by the evaporation. As time reference, we use $t_{\text {ref}}=\phi \ell _{\text {ref}}/E=\phi D/E^2$, which corresponds to the natural time scale for evaporative transport inside the porous medium. As reference concentration and density, we use the initial concentration and density; $X_{\text {ref}}=X_0, \rho _{\text {ref}}=\rho _0\gamma X_0$. Note that the salt mass fraction is itself non-dimensional, hence the new variable ${\hat{X}}$ is only a scaled version of X. For density-driven instability problems, there is usually a prescribed density difference, which is used as the reference density. However, this is not the case here due to the no-flux boundary condition used for salt. Since the salt concentration, and hence, the density could grow unbounded, this model problem does not exhibit a natural density difference. Hence, the initial density scaled with $\gamma$ and $X_0$ is used as reference density. We have here chosen to include also $\gamma$ and $X_{0}$ in the reference density for convenience. This choice is not essential for the non-dimensionalization and not including these two factors would essentially give the same results in the linear stability analysis, just accordingly scaled. The reference velocity is set as a gravitational velocity $Q_{\text {ref}}= {\rho _{\text {ref}}g}K/\mu$, and the reference pressure is chosen to balance the velocity in Darcy’s law $P_{\text {ref}}=\mu \ell _{\text {ref}} Q_{\text {ref}}/K$. To summarize all choices, we list them below:

$$\begin{aligned} \ell _\text {ref} =&\frac{D}{E},\quad t_\text {ref}=\frac{\phi \ell _\text {ref}}{E},\quad X_\text {ref}=X_0,\end{aligned}$$

(23)

$$\begin{aligned} \rho _\text {ref}=&\rho _0\gamma X_0,\quad Q_\text {ref}=\frac{\rho _\text {ref}gK}{\mu },\quad P_\text {ref}=\frac{\mu \ell _\text {ref}Q_\text {ref}}{K} \end{aligned}$$

(24)

We finally introduce the evaporative Rayleigh number $R = Q_{\text {ref}}/E$. Hence, the Rayleigh number describes the ratio between the gravitational flow and the flow induced by the evaporative flux at the top boundary. Note that a larger evaporation rate corresponds to a smaller Rayleigh number. Summarizing, we have

$$\begin{aligned} R=\frac{Q_{\text {ref}}}{E} \text { where } Q_{\text {ref}}=\frac{\gamma \rho _{0}gX_0K}{\mu }. \end{aligned}$$

(25)

Remark 1

In many density-driven instability problems, a typical density difference is used in the reference velocity $Q_{\text {ref}}$ and hence in the Rayleigh number R (see e.g., Riaz et al. 2006). However, in our model setup no typical density difference appears as we do not prescribe a fixed density at the top of the domain. Instead, we propose a reference velocity that only involves the initial density $\rho _0$. The choice of reference velocity does not affect the linear stability analysis results.

Remark 2

Note that the definition of the Rayleigh number does not include the diffusion coefficient D. Here it appears in the reference length, and hence also in the reference time. As we will see later, the critical Rayleigh number will be a function of non-dimensional time and a non-dimensional length parameter, which indirectly gives the dependence of the onset of instabilities on the diffusion.

The non-dimensional model equations are then

$$\begin{aligned} {\hat{\nabla }}\cdot \hat{{\mathbf {Q}}}&= 0, \end{aligned}$$

(26)

$$\begin{aligned} \hat{{\mathbf {Q}}}&= -{\hat{\nabla }} {\hat{P}}+{\hat{\rho }}({\hat{X}}){\mathbf {e}}_z, \end{aligned}$$

(27)

$$\begin{aligned} \partial _{{\hat{t}}}{\hat{X}}&= {\hat{\nabla }}\cdot (-R\hat{{\mathbf {Q}}}{\hat{X}}+{\hat{\nabla }} {\hat{X}}), \end{aligned}$$

(28)

where the non-dimensional density is given as

$$\begin{aligned} {\hat{\rho }}({\hat{X}}) = \frac{1}{\gamma X_0}+{\hat{X}}-1. \end{aligned}$$

(29)

For the non-dimensional variables, we have the initial condition

$$\begin{aligned} {\hat{X}}|_{{\hat{t}}=0} = 1, \end{aligned}$$

(30)

and boundary conditions at the top boundary

$$\begin{aligned} \hat{{\mathbf {Q}}}|_{{\hat{z}}=0}&= -\frac{1}{R}{\mathbf {e}}_z, \end{aligned}$$

(31)

$$\begin{aligned} ({\hat{X}}+{\hat{\nabla }} {\hat{X}})|_{{\hat{z}}=0}\cdot {\mathbf {e}}_z&= 0. \end{aligned}$$

(32)

In the bounded case, the no-flux boundary conditions are imposed on ${\hat{x}},{\hat{y}}=\pm {\hat{\beta }}$, where ${\hat{\beta }} = {\mathcal {W}} E/D$. Hence

$$\begin{aligned} \hat{{\mathbf {Q}}}\cdot {\mathbf {n}}|_{{\hat{x}},{\hat{y}}=\pm {\hat{\beta }}} = 0,\quad {\hat{\nabla }} {\hat{X}}\cdot {\mathbf {n}}|_{{\hat{x}},{\hat{y}}=\pm {\hat{\beta }}} = 0. \end{aligned}$$

(33)

3.3 Ground State Solution

We will investigate the stability of a particular solution of the system of Eqs. (26)–(28) under the conditions (30)–(32). This solution depends on ${\hat{z}}$ and ${\hat{t}}$. It is called the ground state and is denoted by $\{\hat{{\mathbf {Q}}}^0,{\hat{X}}^0,{\hat{P}}^0 \}$.

For the ground state discharge $\hat{{\mathbf {Q}}}^0({\hat{t}},{\hat{z}})$, it is clear from (26) together with the boundary condition (31) that the only possible solution is

$$\begin{aligned} \hat{{\mathbf {Q}}}^0({\hat{t}},{\hat{z}}) = -\frac{1}{R}{\mathbf {e}}_z, \end{aligned}$$

(34)

that is, a constant upward velocity according to the prescribed evaporation rate.

The ground state salt mass fraction ${\hat{X}}^0({\hat{t}},{\hat{z}})$ fulfills the following problem:

$$\begin{aligned} \partial _{{\hat{t}}} {\hat{X}}^0 = \partial _{{\hat{z}}}{\hat{X}}^0 + \partial ^2_{{\hat{z}}}{\hat{X}}^0 \quad {\hat{z}}>0,{\hat{t}}>0, \end{aligned}$$

(35)

$$\begin{aligned} {\hat{X}}^0 +\partial _{{\hat{z}}}{\hat{X}}^0 = 0 \quad {\hat{z}}=0,{\hat{t}}>0,\end{aligned}$$

(36)

$$\begin{aligned} {\hat{X}}^0 = 1 \quad {\hat{z}}>0,{\hat{t}}=0. \end{aligned}$$

(37)

In Appendix A, we show that this problem has the explicit solution

$$\begin{aligned} {\hat{X}}^0({\hat{t}},{\hat{z}}) = 1+\int _0^{{\hat{t}}} \partial _{{\hat{z}}}f(\theta ,{\hat{z}})\ d\theta , \end{aligned}$$

(38)

where

$$\begin{aligned} f(\theta ,{\hat{z}}) = 1-\frac{1}{2} e^{-{\hat{z}}}\text {erfc}\Big (\frac{{\hat{z}}-\theta }{2\sqrt{\theta }} \Big ) - \frac{1}{2}\text {erfc}\Big (\frac{{\hat{z}}+\theta }{2\sqrt{\theta }}\Big ). \end{aligned}$$

(39)

Note in particular that

$$\begin{aligned} \partial _{{\hat{z}}}f(\theta ,{\hat{z}}) = \frac{1}{2}e^{-{\hat{z}}}\text {erfc}\Big (\frac{{\hat{z}}-\theta }{2\sqrt{\theta }}\Big ) + \frac{1}{\sqrt{\pi \theta }}e^{-(\frac{{\hat{z}}+\theta }{2\sqrt{\theta }})^2}, \end{aligned}$$

(40)

enabling a simple evaluation of the integral in (38). The solution is shown in Fig. 3. Clearly, the salt concentration at the top of the domain gradually increases with time and diffuses down through the domain.

Note that since (28) (and hence (35)) does not incorporate the precipitation rate in case the salt mass fraction exceeds the solubility limit, the solution (38) is only valid up to the point where the salt mass fraction at the top reaches the solubility limit. The (non-dimensional) solubility limit depends on the type of salt and the initial concentration. Note, however, that the ground state (38) does not depend on the type of salt. Our analysis can, hence, be straightforwardly applied to any salt type, but only up to times where the corresponding solubility limit is reached.

Fig. 3

Ground state salt mass fractions ${\hat{X}}^0$ (horizontal axis) varying with depth ${\hat{z}}$ (vertical axis) for various times ${\hat{t}}$

The ground state pressure ${\hat{P}}^0({\hat{t}},{\hat{z}})$ is such that

$$\begin{aligned} \partial _{{\hat{z}}}{\hat{P}}^0 = \frac{1}{R} + {\hat{\rho }}({\hat{X}}^0). \end{aligned}$$

(41)

As we only have boundary conditions for the value of the velocity, the pressure is only known up to a constant. Hence,

$$\begin{aligned} {\hat{P}}^0({\hat{t}},{\hat{z}}) = C({\hat{t}})+ (\frac{1}{R}+\frac{1}{\gamma X_0}-1){\hat{z}} + \int _0^{{\hat{z}}} {\hat{X}}^0({\hat{t}},\varsigma )\ d\varsigma , \end{aligned}$$

(42)

where the integration constant $C({\hat{t}})$ cannot be determined, but where a natural choice would be atmospheric pressure at the top boundary. This is, however, not necessary as the following analysis does not depend on the value of the pressure.

3.4 Linear Perturbation and Eigenvalue Problem

Our purpose is to investigate the linear stability of the ground state $\{\hat{{\mathbf {Q}}}^0,{\hat{X}}^0,{\hat{P}}^0\}$. We present the main steps in this section, while details and intermediate steps are given in Appendix B. We write

$$\begin{aligned} \hat{{\mathbf {Q}}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&= \hat{{\mathbf {Q}}}^0({\hat{t}},{\hat{z}})+ {\mathbf {q}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(43)

$$\begin{aligned} {\hat{X}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&={\hat{X}}^0({\hat{t}},{\hat{z}}) + \chi ({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(44)

$$\begin{aligned} {\hat{P}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&={\hat{P}}^0({\hat{t}},{\hat{z}}) + p({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(45)

where ${\mathbf {q}} = (u,v,w), \chi$ and p are small, perturbed quantities, which we will now study further. Note that although these are all non-dimensional, we write them without the hat to simplify the notation. By inserting (43)–(45) into (26)–(28) and into (31)–(32) and linearizing the result, we obtain the linear perturbation equations with corresponding homogeneous boundary conditions for the perturbed quantities. These equations can be expressed using w, $\chi$ and p only.

Since the perturbed quantities fulfill a linear initial-boundary value system, in which none of the coefficients depend on the spatial coordinates ${\hat{x}}$ and ${\hat{y}}$, we consider solutions of the form

$$\begin{aligned} \{w,\chi ,p\}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}) = \{{\tilde{w}},{\tilde{\chi }},{\tilde{p}}\}({\hat{t}},{\hat{z}})\cos ({\hat{a}}_x{\hat{x}})\cos ({\hat{a}}_y{\hat{y}}). \end{aligned}$$

(46)

Here ${\hat{a}}_x$ and ${\hat{a}}_y$ are horizontal wavenumbers. Let ${\hat{a}}^2 = {\hat{a}}_x^2+{\hat{a}}_y^2$. When the domain is unbounded, i.e., the half space $\{{\hat{z}}>0\}$, we allow any ${\hat{a}}>0$. In case of the bounded domain $\{({\hat{x}},{\hat{y}},{\hat{z}}): |{\hat{x}}|,|{\hat{y}}|<{\hat{\beta }},{\hat{z}}>0\}$, we need to choose ${\hat{a}}_x$ and ${\hat{a}}_y$ so that the boundary conditions (33) are satisfied. This requires

$$\begin{aligned} {\hat{a}}_x = n_x\frac{\pi }{{\hat{\beta }}},\quad {\hat{a}}_y = n_y\frac{\pi }{{\hat{\beta }}},\quad n_x,n_y = 1,2,\dots \end{aligned}$$

(47)

Using (46) allows us to eliminate the amplitude ${\tilde{p}}$ as a dependentent variable, and only consider amplitudes ${\tilde{w}}$ and ${\tilde{\chi }}$.

Assuming, in addition, that the coefficients of the initial-boundary value problem do not depend on time ${\hat{t}}$, we can further separate the variables according to

$$\begin{aligned} \{{\tilde{w}},{\tilde{\chi }}\}({\hat{t}},{\hat{z}}) = \{{\hat{w}},{\hat{\chi }}\}({\hat{z}})e^{\sigma {\hat{t}}}, \end{aligned}$$

(48)

see e.g., Subrahmanyan (1961); Nield and Bejan (2017). Here, $\sigma$ is the exponential growth rate in time, while ${\hat{w}}$ and ${\hat{\chi }}$ describe the variability of the perturbation with ${\hat{z}}$. For $\sigma <0$, small perturbations decay in time and the ground state is stable. For $\sigma >0$, perturbations grow in time, and the ground state is unstable.

Since ${\hat{X}}^0 = {\hat{X}}^0({\hat{t}},{\hat{z}})$, the separation of variables as proposed in (48) does not directly apply. This is circumvented by assuming that the rate of change of ${\hat{X}}^0({\hat{t}},{\hat{z}})$ is small compared to any exponentially growing instability. This is known as the quasi steady state approach (QSSA) or the "frozen profile" approach (Riaz et al. 2006; van Duijn et al. 2002). Hence, ${\hat{X}}^0$ is considered to be evaluated at a fixed time.

Thus, we arrive at an eigenvalue problem in terms of $\{{\hat{w}},{\hat{\chi \}}}$ and R, with independent parameters ${\hat{a}}$, ${\hat{t}}$ and $\sigma$. The object is to determine the smallest positive eigenvalue $R=R_*({\hat{a}},{\hat{t}},\sigma )$. We verified numerically, see Appendix C and van Duijn et al. (2019), that there is exchange of stability; i.e., the ground state looses its stability, corresponding to perturbations start growing and $\sigma >0$, for $R>R_*({\hat{a}},{\hat{t}},0)$. Hence, it suffices to analyze the eigenvalue problem for the case of neutral stability; that is, $\sigma =0$. Thus, we need to consider the problem (skipping the asterisk in $R_*)$:

Given ${\hat{a}}>0$, ${\hat{t}}>0$, and ${\hat{X}}^0={\hat{X}}^0({\hat{t}},{\hat{z}})$ by (38), find the smallest $R=R({\hat{a}},{\hat{t}})>0$ such that

$$\begin{aligned} \left. \begin{array}{lr} {\hat{w}}''+{\hat{a}}^2{\hat{\chi }}-{\hat{a}}^2{\hat{w}} = 0 &{} {\hat{z}}>0,\\ {\hat{\chi }}' - R{\hat{w}}\partial _{{\hat{z}}}{\hat{X}}^0 + {\hat{\chi }}'' - {\hat{a}}^2{\hat{\chi }} =0 &{} {\hat{z}}>0,\\ \text {where } {\hat{w}} \text { and } {\hat{\chi }} \text { fulfill } &{} \\ {\hat{w}}=0, {\hat{\chi }}+{\hat{\chi }}'=0 &{} {\hat{z}}=0,\\ {\hat{w}}\rightarrow 0, {\hat{\chi}} \rightarrow 0 &{} {\hat{z}}\rightarrow \infty ,\\ \end{array} \right\} \end{aligned}$$

(49)

has a non-trivial solution.

Note that $'$ denotes the derivative with respect to ${\hat{z}}$.

Remark 3

In the work of Riaz et al. (2006), the ground state results from a diffusion process and depends on ${\hat{\xi }}=\frac{{\hat{z}}}{\sqrt{{\hat{t}}}}$ only. This allows them to perform the coordinate transformation $({\hat{t}},{\hat{z}})\rightarrow ({\hat{t}},{\hat{\xi }})$ and then apply the QSSA to the transformed linear problem. This yields a sharp stability bound. In our case, the ground state results from diffusion and upward convection due to evaporation. Consequently, the ground state (38) does not have such a simple dependence. Therefore, we apply the QSSA directly to the original coordinates $({\hat{t}},{\hat{z}})$.

3.5 Solution of the Eigenvalue Problem

The eigenvalue problem (49) is solved via a Laguerre-Galerkin method. Again, details are given in Appendix B. This results in a system of linear equations for the unknown weights, which can be expressed as a matrix multiplied with a vector containing the weights. Through the eigenvalues of the resulting matrix, we can find the corresponding minimal R as a function of ${\hat{a}}$ for given ${\hat{t}}$, i.e., $R=R({\hat{a}},{\hat{t}})$. Results are shown in Fig. 4. From the figure, we observe that the system becomes gradually more unstable for increasing ${\hat{t}}$. With the Rayleigh number specified by model parameters $R_s=\frac{\gamma \rho _0 gK X_0}{E\mu }$, the system remains stable when $R_s<R$ in Fig. 4. As the curves for R move downward with increasing time, we can for given parameters find a corresponding onset time, which is when $R_s=R$. For a fixed time ${\hat{t}}$, the minimum R always appears for ${\hat{a}}$ approaching 0. This corresponds to longer wavelengths being more unstable. The behavior of R for small wavenumbers ${\hat{a}}$ is detailed in Appendix D. There we also derive an approximation (in fact a lower bound) for the value of R at ${\hat{a}}=0$. We find

$$\begin{aligned} R(0,{\hat{t}}) \ge \frac{2}{{\hat{X}}^0({\hat{t}},0)-1} \text { for all } {\hat{t}}>0. \end{aligned}$$

Hence, the evaporation problem is stable as long as

$$\begin{aligned} R_s<\frac{2}{{\hat{X}}^0({\hat{t}},0)-1}. \end{aligned}$$

Fig. 4

Resulting minimal eigenvalue R (vertical axis) as a function of ${\hat{a}}$ (horizontal axis) for various ${\hat{t}}$

In Fig. 4, we treat ${\hat{a}}$ as a continuous variable. This is allowed when the domain is unbounded. For bounded domains, with zero-flux boundary conditions on the sidewalls, only values of ${\hat{a}}$ that are multiples of $\pi /{\hat{\beta }}$ are allowed (see (47)). We observe in Fig. 4 that for fixed ${\hat{t}}$, the minimum R always occurs for the lowest possible ${\hat{a}}$. Hence in the bounded case, we would only need to solve the eigenvalue problem for ${\hat{a}}=\pi /{\hat{\beta }}$. For a fixed ${\hat{\beta }}$ (and hence a fixed ${\hat{a}}$), the system becomes generally more unstable for later times ${\hat{t}}$. In Fig. 5, we show the critical Rayleigh number for various choices of ${\hat{\beta }}$ as a function of ${\hat{t}}$, which hence are the R-values from Fig. 4 corresponding to ${\hat{a}}=\pi /{\hat{\beta }}$. Note that for larger values of ${\hat{\beta }}$, the critical Rayleigh numbers tend to those for ${\hat{a}}\rightarrow 0$, which corresponds to an infinitely wide domain or an infinitely long wavelength, where no effect of the imposed boundary conditions at the sidewalls are found. These results imply that we can find unique times for onset of instabilities for a given set of parameters, both in the bounded and unbounded case. For a given ${\hat{\beta }}$ and model parameters $R_s$, the corresponding unique onset time can be found in Fig. 5 by finding the corresponding time where $R=R_s$ for our choice of ${\hat{\beta }}$. Similarly, for a given ${\hat{a}}$ and model parameters $R_s$, we can find the corresponding unique onset time in Fig. 4.

Fig. 5

Critical Rayleigh number R (vertical axis) as a function of ${\hat{t}}$ (horizontal axis) for various ${\hat{\beta }}$

Note that although we show results for a large range of non-dimensional times in Figs. 4 and 5, the results are only physically meaningful when the stable salt concentration does not reach its non-dimensional solubility limit. However, the eigenvalue problem (49) can be used for any salt.

3.6 Effect of Varying the Evaporation Rate

Although the solutions of the eigenvalue problem (49) can be used to discuss the effect of all model parameters, we discuss here in particular the effect of the evaporation rate E. The expected influence of E on the stability is a-priori not obvious as it affects the system in two different ways: firstly, a larger evaporation rate corresponds to a stronger vertical, upward throughflow, and secondly, a larger evaporation rate means that the accumulation of salts at the top of the domain is faster. As shown by Homsy and Sherwood (1976); van Duijn et al. (2002), an increased throughflow corresponds to the system being more stable. However, an increased accumulation of salts at the top of the domain is expected to destabilize the system as the density difference is then larger.

To investigate the overall effect of increasing evaporation rate, we recall the definitions

$$\begin{aligned} R=\frac{Q_{\text {ref}}}{E},\quad {\hat{\beta }}=\frac{{\mathcal {W}} E}{D},\quad {\hat{a}}=\frac{\pi }{{\hat{\beta }}}=\frac{\pi D}{{\mathcal {W}} E}, \end{aligned}$$

(50)

where ${\hat{a}}$ is the lowest possible wavenumber in the bounded domain case and hence the most unstable mode. Eliminating E from (50) yields

$$\begin{aligned} R = k {\hat{a}} \quad \text {with } k=\frac{Q_{\text {ref}}{\mathcal {W}}}{D\pi }. \end{aligned}$$

(51)

Recall that ${\mathcal {W}}$ is the half width of the domain, $Q_\text {ref}$ is the reference velocity, and D is the diffusivity of the salt. For a given setup, these numbers would remain constant, but different setups result in different values of k. Returning to Fig. 4, this means that for a given k we look for the points where the curves for fixed times cross the curves $R=k{\hat{a}}$. As a given E corresponds to a given ${\hat{a}}$ (and also R), we can find a corresponding onset time ${\hat{t}}$ for each E. Due to the shape of the curves from Fig. 4, there will be an evaporation rate for which a minimum non-dimensional onset time is found. In Fig. 6, we re-plot Fig. 4 together with the line corresponding to $k=1$, as well as the corresponding onset times as a function of R for various choices of k. The minimum onset times seen in Fig. 6b correspond to the times where the line $R=k{\hat{a}}$ is tangential to a line corresponding to a fixed time in Fig. 6a.

Fig. 6

Relation between onset time and critical Rayleigh number for varying evaporation rate

When the evaporation rate increases, R decreases. Hence, as E increases, we follow a specific curve (corresponding to the chosen value of k) in Fig. 6b, from right to left. We see that as evaporation increases, the corresponding non-dimensional onset time decreases until it reaches a minimum, but then increases as E increases even further. Hence, it appears that for large evaporation rates, increasing E further has a stabilizing effect since the onset time increases. However, Fig. 6b shows non-dimensional onset times. To re-dimensionalize the onset times, we multiply with $t_{\text { ref }} = {\phi D}/{E^2}$. Hence, for increased evaporation rates, the time scaling is smaller. Therefore, when considering the dimensional onset times, these are found to always decrease as the evaporation rate increases. This is independent of the choice of k. Hence, an increased evaporation rate always has a destabilizing effect on the system, since instabilities can appear earlier in dimensional time.

4 Numerical Simulation of the Original Model

We apply a numerical, REV-scale model to simulate the model equations described in Sect. 2. Inline with the formulation in Sect. 2, and due to the applied numerical model using dimensional variables, we now consider the dimensional model. Using dimensional variables also helps to connect back to the physical problem. The numerical model is able to give insights in the distribution and development of the salt concentration in the entire domain. This means that the onset as well as the development of instabilities can be observed in detail. However, it is computationally cost intensive as quite a fine spatial, and temporal discretization is necessary to reduce the influence of numerical diffusion and resolve the instabilities. The numerical model is implemented in the open-source simulator $\mathrm {DuMu}^{x}$ (Koch et al. 2020) for multi-phase, multi-component flow and transport in porous media. It is a research code written in C++ and based on Dune (Bastian et al. 2021), a scientific numerical software framework.

The model considers the precipitation of salt if the solubility limit is exceeded. In this section, we, therefore, first define the reaction term. Additional boundary and initial conditions are specified as we here consider a vertically bounded domain and introduce initial perturbations for the salt mole fraction. Further the spatial and temporal discretizations are described. The domain is discretizied by a cell-centered finite volume scheme, and an implicit Euler method is used for the time discretization. In the end, the evaluation method of the simulation results is described, the numerical onset time is defined, and the influence of different initial perturbations is shown.

4.1 Salt Precipitation Reaction in the Numerical Model

In the numerical model, the precipitation of solid salt is considered if the solubility limit is exceeded. Therefore, the reaction term in the mole balance of the solute and solid salt (Eq. (3) and (8)) has to be specified. For the reaction terms of NaCl $r_\mathrm {solid} = -r^\mathrm {NaCl}$, an equilibrium reaction is assumed. This is based on the assumption that the chemical reaction is so fast that every mol of NaCl above the solubility limit ${\mathsf {x}}^\mathrm {NaCl}_\mathrm {max}$ can precipitate within one numerical time step $\varDelta t$.

$$\begin{aligned} r_\mathrm {solid} = -r^\mathrm {NaCl} = \rho _{\mathrm {mol}} \phi ~ \frac{{\mathsf {x}}^\mathrm {NaCl} - {\mathsf {x}}^\mathrm {NaCl}_\mathrm {max}}{\varDelta t}. \end{aligned}$$

(52)

4.2 Boundary and Initial Conditions

The numerical model uses additional boundary and initial conditions to the ones described in Sect. 2.3. As the domain is finite at the bottom for the numerical simulation, boundary conditions for the bottom are necessary. A Dirichlet boundary condition sets the pressure in the liquid phase corresponding to the hydro-static pressure at the domain depth, and the salt concentration is set equal to the initial mole fraction ${\mathsf {x}}_0^\mathrm {NaCl}$. The depth d of the domain is chosen so that at the time of onset no increase in concentration is observable at the bottom, and so, the influence of the bottom boundary is assumed to be negligible.

$$\begin{aligned} P|_{z=d}&= P_\text {atm} + \rho ({\mathsf {x}}_0^\mathrm {NaCl}) g d, \end{aligned}$$

(53)

$$\begin{aligned} {\mathsf {x}}^\mathrm {NaCl}|_{z=d}&= {\mathsf {x}}_0^\mathrm {NaCl}, \end{aligned}$$

(54)

where $P_\text {atm}$ corresponds to atmospheric pressure.

As initial condition additionally a hydro-static pressure profile is used which leads to better convergence at the beginning of the simulations:

$$\begin{aligned} P|_{t=0} = P_\text {atm} + \rho ({\mathsf {x}}_0^\mathrm {NaCl}) g z. \end{aligned}$$

(55)

The numerical model uses also an initial perturbation for ${\mathsf {x}}^\mathrm {NaCl}$, denoted ${\mathsf {x}}^\mathrm {NaCl}_{0,\mathrm {p}}$, to correspond to the perturbations used in the analytical analysis. If no initial perturbations are applied, instabilities are triggered by tiny numerical errors in the order of machine precision. Instead, two different types of perturbations are used, a periodic and a random one, which can either be applied in the top row of cells or in the whole domain. The periodic perturbation is applied by using a cosine function along the x-coordinate:

$$\begin{aligned} {\mathsf {x}}^\mathrm {NaCl}_{0,\mathrm {p}}(x) ={\mathsf {x}}_\mathrm {0}^\mathrm {NaCl} + A \cdot \mathrm {cos} \left( \frac{2\pi }{\lambda } \cdot x \right) , \end{aligned}$$

(56)

with the wavelength $\lambda$ and the amplitude A. The wavelength and amplitude need to be prescribed. Which amplitude to use will be discussed in Sect. 4.5, while the choice of wavelength is discussed in Sect. 5.1. Alternatively, a random perturbation for ${\mathsf {x}}^\mathrm {NaCl}_{0,\mathrm {p}}$ is used. In this case, values for ${\mathsf {x}}^\mathrm {NaCl}_{0,\mathrm {p}}$ are randomly picked for every discrete cell from a normal distribution ${\mathcal {N}}$ using a mean value of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}={\mathsf {x}}_0^\mathrm {NaCl}$ and a standard deviation $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}$:

$$\begin{aligned} {\mathsf {x}}^\mathrm {NaCl}_{0,\mathrm {p}} \sim {\mathcal {N}} \left( \mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}, \sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^2 \right) . \end{aligned}$$

(57)

The choice of standard deviation is discussed in Sect. 4.5.

4.3 Discretization

For the spatial discretization, a cell-centered finite volume scheme applying the two-point flux approach is used, with a first order upwind scheme for the convective flux and a second order scheme for the diffusive fluxes (Helmig 1997). A first order, implicit Euler method is used for time discretization (Helmig 1997). A convergence study for the spatial and temporal discretization is conducted (see Appendix E), ensuring that there is a negligible influence of numerical diffusion effects. For the spatial discretization, it is important to use finer grid cells than the expected wavelength in order to resolve the instabilities: $\varDelta x \ll \lambda$. Through preliminary testing we found that the expected wavelength depends on the permeability and the vertical density difference and is smaller for higher permeabilities. At least 10 cells are used per wavelength of the highest investigated permeability, as the convergence study shows tolerable errors for this discretization. For longer wavelengths, the error should be even less. Studies on the discretization of $\mathrm {CO_2}$-brine systems in the context of $\mathrm {CO_2}$-storage were done by Elenius and Johannsen (2012). Elenius and Gasda (2021) state also that 10-20 cells per finger are sufficient to resolve the convective flow in the most cases.

As the instabilities are initiated at the top, a fine resolution in z-direction is important near the top boundary ($z=0$). Due to the steep gradient of the salt mole fraction near the top boundary, a smaller $\varDelta z$ better represents these changes and gives also a lower influence of numerical diffusion. Based on the convergence study, a relatively fine $\varDelta z$ is used near the top with $\varDelta z^\mathrm {top} = 3.3 \cdot 10^{-4}~\mathrm {m}$. To lower the computational costs in the lower parts of the domain, coarser cell sizes can be used. Hence, $\varDelta z$ increases continuously toward the bottom.

A time step of $\varDelta t = 50~\mathrm {s}$ is used. This means that the grid velocity of the top cell in z-direction ${\varDelta z^\mathrm {top}}/{\varDelta t}$ is higher than the evaporation rate E and thus is able to capture the evaporation process correctly.

4.4 Evaluation of Numerical Simulations

To estimate the numerical onset time, the mean value $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and the standard deviation $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ of the salt mole fraction ${\mathsf {x}}^\mathrm {NaCl}$ of the grid cells in the top row are calculated. The standard deviation is a measure for the variation of the salt mole fraction, where a standard deviation of zero would correspond to homogeneous salt mole fraction in the top row. Since the development of fingers gradually lead to variations in the salt mole fraction, we expect an increasing standard deviation as the fingers develop. As we start out with a perturbed initial salt mole fraction, the standard deviation will first decrease before it later increases as density instabilities develop. Hence, as a measure for the onset time for the numerical simulations we use the time when the standard deviation is at its minimum. Physically, this corresponds to the case where the convective flux starts to dominate the diffusive fluxes in horizontal direction, which leads to the enhancement of the perturbations. The development of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and their physical interpretation will be discussed in detail in Sect. 5.4.

4.5 Influence of Initial Perturbation on Onset

Figure 7 shows the development of the standard deviation $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and onset time for different parameters for the periodic initial perturbation (Fig. 7a) and the random initial perturbation (Fig. 7b). Also the case without perturbations is shown, where the perturbations are triggered by tiny numerical errors. This figure shows that the amplitude and the standard deviation of the initial perturbation do not affect the onset time. If the initial perturbation is applied only to the top of the domain and not in the whole domain, the onset is later for both perturbation types. However, for the periodic perturbations the difference in onset is relatively small. For the periodic perturbations (56), it is of importance that the width of the domain is a multiple of the initial wavelength. Here, simulations with $\lambda = 0.03$m are used, hence ${\mathcal {W}}=0.30$m is a multiple of it, while ${\mathcal {W}}=0.25$m is not. For the latter case, the onset time is earlier.

In the following investigations, we use a perturbation in the whole domain. This corresponds better to the manner perturbations are applied in the linear stability analysis and hence benefits the comparison of the onset times between the linear stability analysis and the numerical simulations. An amplitude of $A= 10^{-6}$ is used for the periodic perturbations and a standard deviation of $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}= 10^{-6}$ for the random perturbations.

Fig. 7

Influence of different initial perturbations for the numeric simulation on the onset time for periodic initial perturbations (a) and random initial perturbations (b). For the simulations, the parameters listed in Table 2 are used with a permeability of $K=10^{-11}~\mathrm {m^2}$. The parameters amplitude A, half domain width ${\mathcal {W}}$, standard deviation $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}$ are as described in the legend as well as the application area (top or whole). The vertical lines indicate the time of onset for the respective cases. Note that the blue and green vertical lines are in both cases on top of each other

5 Onset and Development of Density Instabilities

Here we compare results from linear stability analysis and numerical simulations with respect to predicted onset times for instabilities and with respect to the behavior of the salt concentration before onset of instabilities. The comparison is kept dimensional, hence results from the linear stability analysis are re-dimensionalized. Most parameters are for simplicity kept fixed and are as specified in Table 2. These parameters are chosen to realistically represent saline water in a porous domain, with evaporation corresponding to 34.7 cm/year. The permeability K of the porous medium is varied. We consider the cases $K=10^{-10}$ m$^2$, $K=10^{-11}$ m$^2$, $K=10^{-12}$ m$^2$ and $K=10^{-13}$ m$^2$.

Table 2 Fixed parameter choices

Note that the specified evaporation rate is used for the numerical simulations as an input parameter. For the linear stability analysis, the choice of evaporation rate translates into a Rayleigh number, which determines the onset time, as described in Sect. 3.5. We separate between the bounded case, where the domain has a fixed width, and the unbounded case, where we investigate the onset of a specific wavelength and use periodic boundary conditions. For both cases, we can compare the onset times from the linear stability analysis and from the numerical simulations.

5.1 Onset Times for a Domain of Fixed Width

We here investigate the onset of instabilities in the bounded case. We use a domain with a fixed width of 60 cm. In the linear stability analysis, we assume that the most unstable wavelength will be dominating, and we find the corresponding onset time for this wavelength, as described in Sect. 3.5. In the numerical simulations, we use an initial random perturbation, which is assumed to trigger the onset of the most unstable wavelength. The development of the standard deviation and the appearing (average) wavelength is shown in Fig. 8. The resulting onset times are in Table 3. Both the analytic and numerical approach show that lower permeabilities correspond to later onset times. Although the numbers are of the same order of magnitude, the deviation between onset times estimated by the linear analysis and numerical simulations is quite large.

Table 3 Onset times from the linear stability analysis for a domain of fixed width. A width of 60 cm has been used for all cases

Fig. 8

Standard deviation and average wavelength (in m) over time for fixed width for $K=10^{\text {-} 10}$ m$^2$ (top left), $K=10^{\text {-} 11}$ m$^2$ (top right), $K=10^{\text {-} 12}$ m$^2$ (bottom left) and $K=10^{\text {-} 13}$ m$^2$ (bottom right). The dotted vertical line indicates time of minimum standard deviation, which is used as the onset time

This analysis reveals some fundamental differences in the underlying assumptions in the two approaches. The linear stability analysis indicates that the most unstable wavelength should be the longest one that will fit into the domain, as lower wavenumbers ${\hat{a}}$ are more unstable, as seen in Fig. 4. For this case that would correspond to a wavelength of 60 cm. In the simulations, we can observe that different wavelengths are dominating before and after the estimated onset time. Since we use a random perturbation, several wavelengths are represented, and they can also interact with each other. The appearing wavelengths after onset are generally found to be shorter for increasing permeability, as seen in Fig. 8. For the lower permeabilities $K=10^{-13}$ m$^2$ and $K=10^{-12}$ m$^2$, the early appearing dominating wavelengths are 15 cm and 12 cm, respectively, which is close to the ones assumed by the linear stability analysis. For the larger permeabilities $K=10^{-11}$ m$^2$ and $K=10^{-10}$ m$^2$, the appearing dominating wavelengths are 4 cm and 1.5 cm, respectively. Although the linear stability analysis indicates that the wavelength of 60 cm should be more unstable and hence preferred, the initial random perturbation have triggered modes that are much shorter in wavelength. The wavelengths that do appear in the numerical simulations depend on the permeability, following a similar trend as observed in Riaz et al. (2006). This is remarkable since the setup is different. Also, using a random perturbation in the numerical simulations can give nonlinear effects as the different wavelengths interact with each other, possibly affecting the resulting onset mode and time. However, the linear stability analysis assumes that the perturbation is a specific wavelength and hence does not account for any interaction between different wavelengths. This motivates to rather use a specific wavelength for the numerical perturbation and compare with the onset time of this wavelength from the linear stability analysis.

5.2 Onset Times of a Fixed Wavelength

We here investigate the onset of instabilities in the unbounded case. Although we use a domain of a width 60 cm for the numerical simulations, we apply periodic boundary conditions on the sidewalls to mimic the domain being unbounded. By using an initial perturbation with a fixed wavelength in the numerical simulations, we investigate the onset of this particular wavelength. This means that the same type of perturbation is used for both numerical simulation and for the linear stability analysis. We use wavelengths based on those that appeared after onset in the numerical simulations in Sect. 5.1, but adjusted such that they fit within the domain. For the linear stability analysis, we then investigate the onset of this particular wavelength, as explained in Sect. 3.5. For the numerical simulations, a small amplitude of the cos-perturbation is used, and the development of the standard deviation and average appearing wavelength is seen in Fig. 9. The used wavelengths and resulting onset times are found in Table 4. For two of the lower permeabilities, no onset time could be found from the numerical simulations. For one case ($K=10^{-12}$ m$^2$ and wavelength 0.12 m), the standard deviation increases throughout the simulation, which means that no local minimum could be detected. For another case ($K=10^{-13}$ m$^2$ and wavelength 0.15 m), salt precipitation occurred before onset of instabilities.

For the cases where numeric onset times could be determined, the onset times generally agree well with the ones predicted by the linear stability analysis. For high permeabilities, the numeric onset times deviate from the analytical ones with less than $40\%$, whereas for low permeabilites, the deviations are less than 10 $\%$. For the $K=10^{-10}$ m$^2$ simulations, we see that the onset of instabilities occurs shortly after the perturbation is applied, possibly because the size of the perturbation was too large, since only a small density increase is needed for fingers to develop (cf. Fig. 11), and hence, triggered the development of the instabilities at an earlier time than expected. In general, we see that lower permeabilities correspond to larger onset times, as also observed by Riaz et al. (2006).

In the numerical simulations, the appearing wavelength at onset is the one used in the initial perturbation. For the permeability $K=10^{-12}$ m$^2$ and using a wavelength of 0.06 m, a slightly longer average wavelength appears shortly after onset as two waves merge. The merging of waves is a common development after onset of instabilities, although it usually appears some time after the instabilities are developed.

Table 4 Onset times from the linear stability analysis and the numerical simulations for specific wavelengths

Fig. 9

Standard deviation and average wavelength (in m) over time for fixed wavelengths for $K=10^{\text {-} 10}$ m$^2$ (top row), $K=10^{\text {-} 11}$ m$^2$ (second row), $K=10^{\text {-} 12}$ m$^2$ (third row) and $K=10^{\text {-} 13}$ m$^2$ (bottom row) for two chosen wavelengths (left and right). The dotted vertical line indicates time of minimum standard deviation, which is used as the onset time. For the case of $K=10^{-13}$ m$^2$ and wavelength 0.15 m (bottom right), time of initial salt precipitation is marked

5.3 Behavior of Top Salt Concentration Before and Near Onset of Instabilities

Using the explicit solution (38) for the ground state salt concentration, we can address the expected development of the salt concentration over time before onset of instabilities. For convenience, the comparison is shown using salt mole fractions, but the numbers could also be converted to salt mass fractions. The largest salt concentration is always found at the top of the domain, hence we focus on this one in the following. Salt precipitates if exceeding ${\mathsf {x}}^\mathrm {NaCl}_\mathrm {max}=0.0977$ (corresponding to $X_{\text {max}}=0.26$). That means, if the ground state salt concentration (38) at the top exceeds this ${\mathsf {x}}^\mathrm {NaCl}_\mathrm {max}$ before onset of instabilities is expected, then salt will instead precipitate. In this case, instabilities will not develop, as the salt mole fraction cannot extend beyond ${\mathsf {x}}^\mathrm {NaCl}_\mathrm {max}$, hence the corresponding density difference is not large enough to trigger instabilities. This occurred for the numerical simulation of $K=10^{-13}$ m$^2$ and using a wavelength of 0.015 m, as seen in Fig. 9g. Note however, as seen from the numerical simulations, the salt concentration at the top of the domain can still increase after the onset of instabilities, before the instabilities are too strong. Hence, one could have salt precipitating after instabilities develop. The linear stability analysis can, however, only determine whether salt precipitation would occur before onset of instabilities.

We compare the salt mole fraction found from the explicit solution (38) with the one from the numerical simulations. For the numerical simulations, we take the average over all the cells in the top row. Since DuMu$^\text {x}$ uses a cell-centered scheme, this means that we are not looking at the salt mass fraction at the top, but $\frac{1}{2}\varDelta z_{\text {top}}$ away from the top. Hence, the explicit solution is, therefore, also evaluated at the height corresponding to the center of the top grid cells.

The time evolution of the salt mole fractions are found in Fig. 10 for the bounded cases from Sect. 5.1, and in Fig. 11 for the unbounded cases from Sect. 5.2. Since the development of the salt mole fraction is not varying with the applied perturbation, we only show the cases corresponding to wavelengths 0.01 m, 0.03 m, 0.06 m and 0.3 m for the permeabilities $K=10^{-10}$ m$^2$, $K=10^{-11}$ m$^2$, $K=10^{-12}$ m$^2$ and $K=10^{-13}$ m$^2$, respectively. The salt mole fractions coming from numerical simulations and explicit solutions are expected to coincide until onset of instabilities. The explicit solutions are plotted beyond the corresponding onset time for comparison, but develop as if instabilities do not occur. Hence, the numerical and explicit solutions should deviate after onset of instabilities. We see, however, that especially for low permeabilities, the explicit and numerical salt mole fractions deviate also slightly before onset of instabilities. For the low permeabilities, the salt mole fraction increases much more before onset of instabilities, compared to the high-permeability cases. This also causes a larger change in the density at the top of the domain, which disrupts the Bousinessq approximation used to derive the explicit solution. However, the overall fit between the explicit solution and the numerical solution is very good. The explicit solution deviate from the numeric one with less than $2\%$ before the onset time, giving confidence that the simplifications used to derive the explicit ground state solution (i.e., neglecting the water mole fraction and applying the Boussinesq approximation) were applicable. We also see clearly that the simulated salt concentration at the top follows to a large extent the analytic solution also after onset of instabilities. This is due to the instabilities being so weak in the beginning, hence their capability to transport salt downward is not developed.

Fig. 10

Salt mole fraction ${\mathsf {x}}^\mathrm {NaCl}$ at top of the domain as a function of time (in seconds), for $K=10^{-10}$ m$^2$ (top left), $K=10^{-11}$ m$^2$ (top right), $K=10^{-12}$ m$^2$ (bottom left), $K=10^{-13}$ m$^2$ (bottom right), when a specific width is used. The estimated onset times are marked as vertical lines

Fig. 11

Salt mole fraction ${\mathsf {x}}^\mathrm {NaCl}$ at top of the domain as a function of time (in seconds), for $K=10^{-10}$ m$^2$ (top left), $K=10^{-11}$ m$^2$ (top right), $K=10^{-12}$ m$^2$ (bottom left), $K=10^{-13}$ m$^2$ (bottom right), when a specific wavelength is analyzed. The estimated onset times are marked as vertical lines. For $K=10^{-11}$ m$^2$ and $K=10^{-13}$ m$^2$, the vertical lines for onset are almost on top of each other

5.4 Development of the Salt Concentration After Onset of Instabilities

In this section, the development of instabilities before and after their onset is discussed, using the results of the numerical simulations. Here we also show the case of salt precipitation which occurs for some parameter sets. Different phases of the development are defined with help of the mean value $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and the standard deviation $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ of the salt mole fraction ${\mathsf {x}}^\mathrm {NaCl}$ of the grid cells in the top row. These phases can be explained and distinguished by the different dominant physical processes.

In Fig. 12, the evaluation of the numeric simulations is shown for the different permeabilities and initial perturbations. We here present only one wavelength (0.01 m, 0.03 m, 0.06 m or 0.3 m) per permeability for the periodic initial perturbations as they show a similar general behavior. The development of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ and $\sigma _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ over time indicates the different phases of the formation of instabilities.

In the first phase, the initial standard deviation decreases due to the spreading of the initially applied perturbations by molecular diffusion. In Fig. 13a, the influence of the different fluxes on the resulting flux is shown for the first phase. As already mentioned in Sect. 2.2, the higher density at the location of the perturbation leads to an increased gravitational force and a slowed down convective upward flux. As we apply a constant evaporation rate a lower pressure develops at these locations. This horizontal pressure gradient induces a horizontal component to the convective flux toward the perturbation. However, in the first phase the diffusive flux dominates the convective flux in the horizontal direction, which leads to a degradation of the perturbation. In the vertical direction, the driving force of the pressure gradient outweighs the gravitational force. This results in an upward convective flux which also dominates the diffusive flux in the vertical direction. This leads to an upward transportation and accumulation of salt at the top during this phase.

The second phase starts at the time of onset, when the standard deviation reaches its minimum and the perturbations start to increase. The reason for that can be seen in Fig. 13b. The increasing salt concentration and fluid density enhances the gravitational downward forces, which in the following enhances the horizontal component of the convective flux toward the perturbation. In this phase, the convective flux dominates the diffusive flux in horizontal direction. This leads to an increased transport of salt toward the perturbation and consequently to its enhancement. The vertical direction of the resulting flux is still upward, which increases the salt concentration at the top. This is also demonstrated by the continuous increase of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ during this phase.

For the higher permeabilities $K=10^{-10} - 10^{-12}~\mathrm {m^2}$, the third phase is characterized by a resulting downward flow and starts at the maximum value of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$. Until the start of the third phase, the liquid density has increased so much at the top and especially at the location of perturbations that the high gravitational force causes an convective and resulting flux downward; so-called fingers. With a lower permeability a higher density difference is needed to overcome the resistance of the porous medium, resulting in higher maximum values for $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$. The resulting flux transports the accumulated salt at the top downward, which leads to a decrease of $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$. Later in this phase, $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ stabilizes as the upward transported salt equals the amount which is transported downward with the fingers. A larger value for this stabilized salt mole fraction is observed for lower permeabilities. To match the upward flow, determined by the constant evaporation rate, the downward flow needs larger density differences for lower permeabilities. In all these cases, the solubility limit of the salt is never reached, and thus, no salt precipitation is observed in these systems. For the lowest considered permeability $K = 10^{-13}~\mathrm {m^2}$, $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ reaches the solubility limit before a convective downward flow develops. Here salt precipitates and $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ stays constant at the solubility limit as we use an equilibrium approach to simulate the precipitation reaction (see Eq. (52)).

Note that in case of no precipitation, further phases for the instabilities can be defined. The fingers start to merge and form larger fingers with longer wavelengths. As we concentrate on the initial development of the instabilities, we refer to Slim (2014) for a detailed description of these merging regimes. Slim describes similar phases, although there the instabilities are not evaporation-driven.

Fig. 12

Result from the numerical simulations for a domain of fixed width (left) and for fixed wavelengths (right) and the different permeabilities. For each case, the development of the mean value and standard deviation of the salt concentration is shown, as well as different phases of the development of instabilities

Fig. 13

Important processes and forces in the different phases of instability development. Left the convective streamlines and velocity arrows as well as the salt mole fraction in the background are shown exemplary for one perturbation for $K=10^{-11}$ m$^2$ and periodic initial perturbations. Right a schematic overview of the fluxes and forces describes the formation of the resulting flux

6 Final Remarks

As water evaporates from a porous medium saturated with saline water, the accumulated salt near the top boundary will either trigger density instabilities, or precipitate in form of a salt crust, or both. In this work, we have addressed the onset of instabilities using two approaches: analytically by applying a linear stability analysis and numerically by performing simulations. The linear stability analysis simplifies the governing equations and formulates an eigenvalue problem giving conditions for whether and when instabilities can develop. The advantage of the linear stability analysis is that results for a large range of parameters can be obtained at very low costs. The numerical simulations can address the original governing equations and can time-step these to address when instabilities develop. The computational costs are larger, but also information for the further development after the onset of instabilities can be obtained.

The onset times of instabilities depend not only on the physical parameters as the medium’s permeability and the strength of the evaporation rate, but also on which type of instability is considered. The boundary conditions on the sidewalls and which wavelength develops, affect the development of the instabilities. We here considered two cases; either a bounded case with no-flux boundary conditions on the sidewalls, where we tried to trigger the most unstable wavelength, or an unbounded case using periodic boundary conditions, where we tried to trigger a specific wavelength. For the first case, the onset times predicted by the two approaches deviate as the applied perturbation is different, and the two methods predict the onset of different instability modes. For the second case, the onset times largely coincide. In both cases, the development of the salt concentration up to onset of instabilities match up to the difference arising from using the Boussinesq approximation in the analytic case. This gives confidence that the two methods can correctly predict the onset of instabilities, when a specific wavelength is expected.

The numerical experiments show the development of the salt concentration also after onset of instabilities. In particular, we see how the salt concentration at the top of the domain continues to increase for some time after onset, as the instabilities are in the beginning too weak to cause a net downward transport of salt. This means that salt can still precipitate even if instabilities have been triggered.

The linear stability analysis can quickly give criteria for onset of instabilities for a large range of parameters. The numerical simulations can further give detailed information of the further development of instabilities, when applying given parameter choices. From the linear stability analysis, we see how the onset of instabilities depend strongly on parameters such the strength of the evaporation rate and of the permeability, where the latter was also investigated by the numerical simulations. The times for onset of instabilities are found to be in the range of hours to days for a realistic evaporation rate, depending on the permeability. This means that, in the lack of rainfall in that period, our findings give onset times that are realistic especially in arid regions. However, for specific implications, field-related analysis is necessary. This study shows that the current framework is suitable as analysis strategy for onset times of evaporation-induced density instabilities.

Our analysis opens also for comparison with column experiments that consider evaporation from the top of a porous column saturated with saline water having different salts, e.g., Piotrowski et al. (2020). However, the current analysis is performed under the assumption that the porous medium remains fully saturated. Hence, an extension to unsaturated porous media would give more accurate results, also in the context of relating to field observations. To accommodate such an extension, Richards equation for the evolution of the water saturation needs to be included—both in the linear stability analysis and in the numerical simulations. In this case, also capillary forces play a role for the evolution of the water saturation, giving potentially more interactions between evaporation and subsequent density instabilities. The current study remains valid for the case when the capillary pressure stays below the entry pressure, while further research is needed to address the case of varying water saturation.

Appendices

A Explicit Solution of the Salt Ground State

For the (non-dimensional) ground-state salt concentration ${\hat{X}}^0$, which is needed in Sect. 3.3, we here derive the explicit solution of

$$\begin{aligned} \partial _{{\hat{t}}} {\hat{X}}^0 = \partial _{{\hat{z}}}{\hat{X}}^0 + \partial ^2_{{\hat{z}}}{\hat{X}}^0 \quad {\hat{z}}>0,{\hat{t}}>0, \\ {\hat{X}}^0 +\partial _{{\hat{z}}}{\hat{X}}^0 = 0 \quad {\hat{z}}=0,{\hat{t}}>0,\\ {\hat{X}}^0 = 1 \quad {\hat{z}}>0,{\hat{t}}=0. \end{aligned}$$

We rewrite this problem in terms of the flux; $f = {\hat{X}}^0 +\partial _{{\hat{z}}}{\hat{X}}^0$. Since $\partial _{{\hat{t}}} {\hat{X}}^0 = \partial _{{\hat{z}}} f$, we have

$$\begin{aligned} \partial _{{\hat{t}}} f = \partial _{{\hat{t}}}{\hat{X}}^0 + \partial _{{\hat{t}} {\hat{z}}}{\hat{X}}^0 = \partial _{{\hat{z}}} f+\partial _{{\hat{z}}}^2 f \quad {\hat{z}}>0,{\hat{t}}>0, \\ f = 0 \quad {\hat{z}}=0,{\hat{t}}>0,\\ f = 1 \quad {\hat{z}}>0,{\hat{t}}=0. \end{aligned}$$

Writing instead in terms of $g=1-f$, we have

$$\begin{aligned} \partial _{{\hat{t}}} g = \partial _{{\hat{z}}} g+\partial _{{\hat{z}}}^2 g \quad {\hat{z}}>0,{\hat{t}}>0, \\ g = 1 \quad {\hat{z}}=0,{\hat{t}}>0,\\ g = 0 \quad {\hat{z}}>0,{\hat{t}}=0, \end{aligned}$$

which has a known explicit solution (Bear 1972), namely

$$\begin{aligned} g({\hat{t}},{\hat{z}}) = \frac{1}{2}e^{-{\hat{z}}}\text {erfc}\Big (\frac{{\hat{z}}-{\hat{t}}}{2\sqrt{{\hat{t}}}}\Big )+\frac{1}{2}\text {erfc}\Big (\frac{{\hat{z}}+{\hat{t}}}{2\sqrt{{\hat{t}}}}\Big ). \end{aligned}$$

Hence,

$$\begin{aligned} f({\hat{t}},{\hat{z}}) = 1-\frac{1}{2}e^{-{\hat{z}}}\text {erfc}\Big (\frac{{\hat{z}}-{\hat{t}}}{2\sqrt{{\hat{t}}}}\Big )-\frac{1}{2}\text {erfc}\Big (\frac{{\hat{z}}+{\hat{t}}}{2\sqrt{{\hat{t}}}}\Big ). \end{aligned}$$

Using again that $\partial _{{\hat{t}}} {\hat{X}}^0 = \partial _{{\hat{z}}} f$, we obtain

$$\begin{aligned} {\hat{X}}^0({\hat{t}},{\hat{z}}) = 1+\int _0^{{\hat{t}}} \partial _{{\hat{z}}}f(\theta ,{\hat{z}})\ d\theta . \end{aligned}$$

B Derivation of and Solution Strategy for the Eigenvalue Problem

We here go detailed through the steps for deriving the eigenvalue problem (49). To investigate the stability of the ground state $\{\hat{{\mathbf {Q}}}^0,{\hat{X}}^0,{\hat{P}}^0\}$ from Sect. 3.3, we write

$$\begin{aligned} \hat{{\mathbf {Q}}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&= \hat{{\mathbf {Q}}}^0({\hat{t}},{\hat{z}})+ {\mathbf {q}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(58)

$$\begin{aligned} {\hat{X}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&={\hat{X}}^0({\hat{t}},{\hat{z}}) + \chi ({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(59)

$$\begin{aligned} {\hat{P}}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&={\hat{P}}^0({\hat{t}},{\hat{z}}) + p({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}), \end{aligned}$$

(60)

where ${\mathbf {q}} = (u,v,w), \chi$ and p are small, perturbed quantities. Since $\hat{{\mathbf {Q}}},{\hat{X}}$ and ${\hat{P}}$ still need to solve the original equations and boundary conditions, we achieve equations and boundary conditions for the perturbed quantities. Inserting (58)–(60) into (26)–(28) and into (31)–(32) and linearizing, we obtain the linear perturbation equations

$$\begin{aligned} {\hat{\nabla }}\cdot {\mathbf {q}}&= 0, \end{aligned}$$

(61)

$$\begin{aligned} {\mathbf {q}}&= -{\hat{\nabla }} p + \chi {\mathbf {e}}_z, \end{aligned}$$

(62)

$$\begin{aligned} \partial _{{\hat{t}}}\chi&= \partial _{{\hat{z}}}\chi - Rw \partial _{{\hat{z}}}{\hat{X}}^{0}+{\hat{\nabla }}^{2}\chi , \end{aligned}$$

(63)

with boundary conditions at the top

$$\begin{aligned} {\mathbf {q}}|_{{\hat{z}}=0}&= {\mathbf {0}}, \end{aligned}$$

(64)

$$\begin{aligned} (\chi +\partial _{{\hat{z}}}\chi )|_{{\hat{z}}=0}&= 0. \end{aligned}$$

(65)

Equations (61)–(63) can be written in terms of $\{w,\chi ,p\}$ only, see e.g., van Duijn et al. (2019). This results in

$$\begin{aligned} {\hat{\nabla }}^{2}w&= (\partial _{{\hat{x}}}^2+\partial _{{\hat{y}}}^2)\chi \end{aligned}$$

(66)

$$\begin{aligned} \partial _{{\hat{z}}} w&= (\partial _{{\hat{x}}}^2+\partial _{{\hat{y}}}^2)p \end{aligned}$$

(67)

$$\begin{aligned} \partial _{{\hat{t}}}\chi&= \partial _{{\hat{z}}}\chi - Rw\partial _{{\hat{z}}}{\hat{X}}^{0}+{\hat{\nabla }}^2\chi , \end{aligned}$$

(68)

with boundary conditions

$$\begin{aligned} w|_{{\hat{z}}}=0&= 0, \end{aligned}$$

(69)

$$\begin{aligned} (\chi +\partial _{{\hat{z}}}\chi )|_{{\hat{z}}=0}&= 0. \end{aligned}$$

(70)

We seek solutions of this system satisfying

$$\begin{aligned} w,\chi \rightarrow 0 \text { as } {\hat{z}}\rightarrow \infty . \end{aligned}$$

(71)

Since (66)–(71) is a linear initial-boundary problem with no coefficients depending on the spatial coordinates ${\hat{x}}$ and ${\hat{y}}$, we consider solutions of the form

$$\begin{aligned} \{w,\chi ,p\}({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}}) = \{{\tilde{w}},{\tilde{\chi }},{\tilde{p}}\}({\hat{t}},{\hat{z}})\cos ({\hat{a}}_x{\hat{x}})\cos ({\hat{a}}_y{\hat{y}}). \end{aligned}$$

(72)

Here ${\hat{a}}_x$ and ${\hat{a}}_y$ are horizontal wavenumbers. Substituting (72) into (66)–(68) yields for the amplitudes $\{{\tilde{w}},{\tilde{\chi }},{\tilde{p}}\}$

$$\begin{aligned} \partial _{{\hat{z}}}^2{\tilde{w}} -{\hat{a}}^2{\tilde{w}}&=-{\hat{a}}^2{\tilde{\chi }}, \end{aligned}$$

(73)

$$\begin{aligned} \partial _{{\hat{z}}}{\tilde{w}}&= -{\hat{a}}^2{\tilde{p}},\end{aligned}$$

(74)

$$\begin{aligned} \partial _{{\hat{t}}}{\tilde{\chi }}&= \partial _{{\hat{z}}}{\tilde{\chi }} - R{\tilde{w}}\partial _{{\hat{z}}}{\hat{X}}^0 + \partial _{{\hat{z}}}^2{\tilde{\chi }} - {\hat{a}}^2{\tilde{\chi }}, \end{aligned}$$

(75)

where ${\hat{a}}^2={\hat{a}}_x^2+{\hat{a}}_y^2$. Expressions for the horizontal discharge components follow from Darcy’s law:

$$\begin{aligned} u({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&= -\partial _{{\hat{x}}}p = -\frac{{\hat{a}}_x}{{\hat{a}}^2}\partial _{{\hat{z}}}{\tilde{w}}\sin ({\hat{a}}_x{\hat{x}})\cos ({\hat{a}}_y{\hat{y}}),\end{aligned}$$

(76)

$$\begin{aligned} v({\hat{t}},{\hat{x}},{\hat{y}},{\hat{z}})&= -\frac{{\hat{a}}_y}{{\hat{a}}^2}\partial _{{\hat{z}}}{\tilde{w}}\cos ({\hat{a}}_x{\hat{x}})\sin ({\hat{a}}_y {\hat{y}}). \end{aligned}$$

(77)

When the domain is unbounded, i.e., the half space $\{{\hat{z}}>0\}$, we consider (73)–(75), subject to (69)–(70) for any ${\hat{a}}>0$. In case of the bounded domain $\{({\hat{x}},{\hat{y}},{\hat{z}}): |{\hat{x}}|,|{\hat{y}}|<{\hat{\beta }},{\hat{z}}>0\}$ we need to choose ${\hat{a}}_x$ and ${\hat{a}}_y$ so that the boundary conditions (33) are satisfied. This requires

$$\begin{aligned} {\hat{a}}_x = n_x\frac{\pi }{{\hat{\beta }}},\quad {\hat{a}}_y = n_y\frac{\pi }{{\hat{\beta }}},\quad n_x,n_y = 1,2,\dots \end{aligned}$$

(78)

A sketch of a horizontal discharge field is given in Fig. 14. In this figure, we have used ${\hat{\beta }}=1$ as well as ${\hat{a}}_x={\hat{a}}_y=\pi$, corresponding to one full oscillation in both ${\hat{x}}$ and ${\hat{y}}$ direction.

Fig. 14

Horizontal discharge (u, v) seen from above for a cross section $\{({\hat{x}},{\hat{y}}): |{\hat{x}}|,|{\hat{y}}|<{\hat{\beta \}}}$, here using ${\hat{\beta }}=1$

It is clear that it suffices to consider only Eqs. (73) and (75), subject to boundary conditions (69)–(71). Once they are solved, the pressure results from (74) and the horizontal discharge from (76) to (77).

Using the linearity of (73) and (75), and assuming that the coefficients do not depend on time t, we can further separate the variables according to

$$\begin{aligned} \{{\tilde{w}},{\tilde{\chi }}\}({\hat{t}},{\hat{z}}) = \{{\hat{w}},{\hat{\chi }}\}({\hat{z}})e^{\sigma {\hat{t}}}, \end{aligned}$$

(79)

where $\sigma$ is the exponential growth rate in time, while ${\hat{w}}$ and ${\hat{\chi }}$ describe the variability of the perturbation with ${\hat{z}}$.

Since ${\hat{X}}^0 = {\hat{X}}^0({\hat{t}},{\hat{z}})$, the separation of variables as proposed in (79) does not directly apply. As already explained in Sect. 3.4, we circumvent this by applying the quasi steady state approach (QSSA) or the "frozen profile" approach: For any fixed ${\hat{t}}_*>0$, we consider ${\hat{\tau }} = {\hat{t}}-{\hat{t}}_*$ as the new time variable. Then (75) becomes

$$\begin{aligned} \partial _{{\hat{\tau }}}{\tilde{\chi }} = \partial _{{\hat{z}}}{\tilde{\chi }} - R{\tilde{w}}\partial _{{\hat{z}}}{\hat{X}}^0({\hat{t}}_*+{\hat{\tau }},{\hat{z}}) + \partial _{{\hat{z}}}^2{\tilde{\chi }} - {\hat{a}}^2{\tilde{\chi }}. \end{aligned}$$

(80)

In this equation, we take ${\hat{\tau }}$ small, $0<{\hat{\tau \ll }}1$, and write ${\hat{X}}^0({\hat{t}}_*+{\hat{\tau }},{\hat{z}})\approx {\hat{X}}^0({\hat{t}}_*,{\hat{z}})$, i.e., the frozen profile. Setting now

$$\begin{aligned} \{{\tilde{w}},{\tilde{\chi }}\}({\hat{\tau }},{\hat{z}}) = \{{\hat{w}},{\hat{\chi }}\}({\hat{z}})e^{\sigma {\hat{\tau }}}, \end{aligned}$$

(81)

we obtain

$$\begin{aligned} \partial _{{\hat{z}}}^2 w-{\hat{a}}^2{\hat{w}}&=-{\hat{a}}^2{\hat{\chi }} \end{aligned}$$

(82)

$$\begin{aligned} \sigma {\hat{\chi }}&= \partial _{{\hat{z}}}{\hat{\chi }} - R{\hat{w}}\partial _{{\hat{z}}}{\hat{X}}^0({\hat{t}}_*,{\hat{z}}) + \partial _{{\hat{z}}}^2{\hat{\chi }} - {\hat{a}}^2{\hat{\chi }}, \end{aligned}$$

(83)

for $0<{\hat{z}}<\infty$ subject to (69)–(71).

For given ${\hat{a}}>0,{\hat{t}}_*>0$ and $\sigma \in {\mathbb {R}}$, this is an eigenvalue problem in terms of $\{{\hat{w}},{\hat{\chi \}}}$ and R. The object is to determine the smallest positive eigenvalue $R=R_*({\hat{a}},{\hat{t}}_*,\sigma )$. As shown in Appendix C, there is exchange of stability; i.e.,

$$\begin{aligned} R_*({\hat{t}},{\hat{t}}_*,\sigma ) \gtrless R_*({\hat{a}},{\hat{t}}_*,0) \text { if and only if } \sigma \gtrless 0. \end{aligned}$$

(84)

This means that if the parameters of the problem are such that $R>R_*({\hat{a}},{\hat{t}}_*,0)$, then at time ${\hat{t}}_*$ a perturbation with wavenumber ${\hat{a}}$ will emerge, implying that the ground state looses its stability at time ${\hat{t}}_*$.

Based on this observation, it suffices to analyze the eigenvalue problem for the case of neutral stability; that is, $\sigma =0$. Thus, we need to consider the problem (dropping the asterisk in ${\hat{t}}_*$ and $R_*$):

Given ${\hat{a}}>0$, ${\hat{t}}>0$, and ${\hat{X}}^0={\hat{X}}^0({\hat{t}},{\hat{z}})$ by (38), find the smallest $R=R({\hat{a}},{\hat{t}})>0$ such that

$$\begin{aligned} \left. \begin{array}{lr} {\hat{w}}''+{\hat{a}}^2{\hat{\chi }}-{\hat{a}}^2{\hat{w}} = 0 &{} {\hat{z}}>0,\\ {\hat{\chi }}' - R{\hat{w}}\partial _{{\hat{z}}}{\hat{X}}^0 + {\hat{\chi }}'' - {\hat{a}}^2{\hat{\chi }} =0 &{} {\hat{z}}>0,\\ \text {where } {\hat{w}} \text { and } {\hat{\chi }} \text { fulfill } &{} \\ {\hat{w}}=0, {\hat{\chi }}+{\hat{\chi }}'=0 &{} {\hat{z}}=0,\\ {\hat{w}}\rightarrow 0, {\hat{\chi}} \rightarrow 0 &{} {\hat{z}}\rightarrow \infty ,\\ \end{array} \right\} \end{aligned}$$

(85)

has a non-trivial solution. This is the same as (49).

The eigenvalue problem (85) is solved via a Laguerre-Galerkin method. We let

$$\begin{aligned} {\hat{\chi }}({\hat{z}}) = \sum _{n=0}^\infty {\hat{\chi}} _n\zeta _n({\hat{z}}), \quad {\hat{w}}({\hat{z}}) = \sum _{n=0}^\infty {\hat{w}}_n\eta _n({\hat{z}}), \end{aligned}$$

(86)

where the basis functions are given by

$$\begin{aligned} \zeta _n({\hat{z}}) = e^{-\frac{{\hat{z}}}{2}}\Big ({\mathcal {L}}_n({\hat{z}})+\frac{1-2n}{1+2n}{\mathcal {L}}_{n+1}({\hat{z}})\Big ) \text { and } \eta _n({\hat{z}}) = e^{-\frac{{\hat{z}}}{2}}\Big ({\mathcal {L}}_n({\hat{z}})-{\mathcal {L}}_{n+1}({\hat{z}})\Big ), \end{aligned}$$

(87)

and where ${\mathcal {L}}_n$ is the Laguerre polynomial of degree n  (Temme 1996); i.e., 

$$\begin{aligned} {\mathcal {L}}_n({\hat{z}}) = \sum _{\ell =0}^n(-1)^\ell \begin{pmatrix} n\\ \ell \end{pmatrix} \frac{{\hat{z}}^\ell }{\ell !} \text { with } {\mathcal {L}}_n(0)=1 \text { and } {\mathcal {L}}'_n(0)=-n. \end{aligned}$$

(88)

The special combinations in (87) are chosen so that that $\zeta _n({\hat{z}})$ and $\eta _n({\hat{z}})$ satisfy the boundary conditions for ${\hat{\chi }}$ and ${\hat{w}}$ in the fourth and fifth line of (85), respectively. Inserting (87) into the first two lines of (85), multiplying with $\zeta _m$ and $\eta _m$ and integrating with respect to ${\hat{z}}$ yields

$$\begin{aligned}&\sum _{n=0}^{\infty }{\hat{w}}_n\int _0^\infty \eta _n''\eta _m d{\hat{z}} + {\hat{a}}^2\sum _{n=0}^\infty {\hat{\chi }}_n\int _0^\infty \zeta _n\eta _m d{\hat{z}} -{\hat{a}}^2\sum _{n=0}^\infty {\hat{w}}_n\int _0^\infty \eta _n\eta _m d{\hat{z}} = 0,\\&\quad \sum _{n=0}^\infty {\hat{\chi }}_n\int _{0}^\infty \zeta _n'\zeta _m d{\hat{z}} - R\sum _{n=0}^\infty {\hat{w}}_n\int _{0}^\infty \partial _{{\hat{z}}} {\hat{X}}^0\eta _n\zeta _m d{\hat{z}} \\&\quad +\sum _{n=0}^\infty {\hat{\chi }}_n\int _{0}^\infty \zeta _n''\zeta _m d{\hat{z}} - {\hat{a}}^2\sum _{n=0}^\infty {\hat{\chi }}_n\int _{0}^\infty \zeta _n\zeta _m d{\hat{z}} =0. \end{aligned}$$

We truncate this expression at a large number $n=N$. Inspecting the convergence of the terms, we found $N=32$ to be sufficient. The integrals can be determined analytically using the properties of Laguerre polynomials. The only exception is the integral involving $\partial _{{\hat{z}}}{\hat{X}}^0$, which is approximated using a Gauss-Laguerre quadrature rule. This results in a system of linear equations for the weights ${\hat{\chi _n}}$ and ${\hat{w}}_n$, which can be expressed as a matrix multiplied with a vector containing the weights. The eigenvalues of this matrix correspond to the eigenvalues of (85).

C Investigation of Exponential Growth Rate

We here include the exponential growth rate $\sigma$ as a parameter in the eigenvalue problem, as considered in Sect. 3.4 and Appendix B. As introduced in (48) and (79), $\sigma >0$ corresponds to perturbations growing in time, while for $\sigma <0$ perturbations decay with time. When keeping $\sigma$ in the perturbed equations, we now obtain the following eigenvalue problem:

Given ${\hat{a}}>0$, ${\hat{t}}>0$, $\sigma \in {\mathbb {R}}$ and ${\hat{X}}^0={\hat{X}}^0({\hat{t}},{\hat{z}})$ by (38), find the smallest $R=R({\hat{a}},{\hat{t}},\sigma )>0$ such that

$$\begin{aligned} \left. \begin{array}{lr} {\hat{w}}''+{\hat{a}}^2{\hat{\chi }}-{\hat{a}}^2{\hat{w}} = 0 &{} {\hat{z}}>0,\\ {\hat{\chi }}' - R{\hat{w}}\partial _{{\hat{z}}}{\hat{X}}^0 + {\hat{\chi }}'' - ({\hat{a}}^2+\sigma ){\hat{\chi }} =0 &{} {\hat{z}}>0,\\ \text {where } {\hat{w}} \text { and } {\hat{\chi }} \text { fulfill } &{} \\ {\hat{w}}=0, {\hat{\chi }}+{\hat{\chi }}'=0 &{} {\hat{z}}=0,\\ {\hat{w}}\rightarrow 0, {\hat{\chi}} \rightarrow 0 &{} {\hat{z}}\rightarrow \infty , \end{array} \right\} \end{aligned}$$

(89)

has a non-trivial solution.

For given ${\hat{a}},{\hat{t}}>0$ and $\sigma \in {\mathbb {R}}$, (89) is an eigenvalue problem where R is to be determined. We follow the same strategy as described in Sect. 3.5 to discretize and solve the eigenvalue problem. The corresponding version of Fig. 4 when including $\sigma$ is shown in Fig. 15.

Fig. 15

Resulting eigenvalue R (vertical axis) as a function of wavenumber ${\hat{a}}$ (horizontal axis) for various ${\hat{t}}$ and $\sigma$. Solid lines correspond to $\sigma =0$, while dashed lines correspond to $\sigma =\pm 0.01$, dotted lines to $\sigma =\pm 0.1$ and dashed-dotted to $\sigma =\pm 1$. The curves lying below the corresponding solid curve are for negative $\sigma$ and the ones above the corresponding solid curve are for positive $\sigma$. Note that the eigenvalue problem degenerates when $({\hat{a}}^2+\sigma )=0$, which is why the lines for negative $\sigma$ do not extend beyond ${\hat{a}}=\sqrt{-\sigma }$

From Fig. 15, we observe the following: For negative $\sigma$ (corresponding to perturbations decaying with time; i.e., stability), we are always below the curves corresponding to $\sigma =0$ when looking at same ${\hat{t}}$ and ${\hat{a}}$. This means, when we are below a solid curve, which means that the Rayleigh number is lower than the one on the vertical axis, we have stability since a perturbation will decay with time. Similarly, for positive $\sigma$ (corresponding to perturbation growing with time), we are always above the curves corresponding to $\sigma =0$ when looking at same ${\hat{t}}$ and ${\hat{a}}$. This means, when we are above a solid curve, which means that the Rayleigh number is larger than the one on the vertical axis, a perturbation will grow exponentially with time. From this, we can conclude that investigating $\sigma =0$ is the relevant case for the eigenvalue problem, as the eigenvalues found by using $\sigma =0$ correspond to the shift from perturbations growing or decaying.

Note that the eigenvalue problem (89) degenerates when ${\hat{a}}^2+\sigma =0$, which is why the curves for negative $\sigma$ are not extended beyond ${\hat{a}}=\sqrt{-\sigma }$. If one would like to investigate very small wavenumbers ${\hat{a}}$, one could overcome this by choosing a correspondingly small $\sigma$ to avoid (or, more correctly, shift) the degeneracy.

D The Behavior for Small Wavenumbers ${\hat{a}}$

Figure 4 shows that for each ${\hat{t}}>0$, the Rayleigh number $R({\hat{a}},{\hat{t}})$ has a finite value as ${\hat{a}} \searrow 0$. This behavior is different from other cases where $R({\hat{a}},{\hat{t}})\rightarrow \infty$ as ${\hat{a}}\searrow 0$ and has a positive minimum at some critical wavenumber ${\hat{a}}_L>0$, e.g., Riaz et al. (2006); van Duijn et al.( 2002). Here we provide an explanation for an approximate eigenvalue problem. We first observe that for each ${\hat{t}}>0$, the ground state solution ${\hat{X}}^0({\hat{t}},{\hat{z}})$ from (38) has its maximum at ${\hat{z}}=0$ and decreases in a convex way toward ${\hat{X}}^0({\hat{t}},\infty )=1$. The idea is to replace ${\hat{X}}^0({\hat{t}},{\hat{z}})$ by the expression

$$\begin{aligned} {\hat{U}}^0({\hat{t}},{\hat{z}}) = 1+e^{-{\hat{z}}}({\hat{X}}^0({\hat{t}},0)-1). \end{aligned}$$

(90)

Figure 16 shows the profiles of ${\hat{X}}^0$ and ${\hat{U}}^0$ for various ${\hat{t}}>0$. Note that they have the same qualitative behavior and that the relative error becomes smaller as ${\hat{t}}$ increases. We propose to use ${\hat{U}}^0({\hat{t}},{\hat{z}})$ as ground state in the eigenvalue problem (49). Setting

$$\begin{aligned} R^* = ({\hat{X}}^0({\hat{t}},0)-1)R, \end{aligned}$$

(91)

we obtain the neutral stability ($\sigma =0$) eigenvalue problem:

Given ${\hat{a}}>0$, find the smallest $R^*=R^*({\hat{a}})>0$ such that

$$\begin{aligned} \left. \begin{array}{lr} {\hat{w}}''+{\hat{a}}^2{\hat{\chi }}-{\hat{a}}^2{\hat{w}} = 0 &{} {\hat{z}}>0,\\ {\hat{\chi }}' + R^*e^{-{\hat{z}}}{\hat{w}} + {\hat{\chi }}'' - ({\hat{a}}^2+\sigma ){\hat{\chi }} =0 &{} {\hat{z}}>0,\\ \text {where } {\hat{w}} \text { and } {\hat{\chi }} \text { fulfill } &{} \\ {\hat{w}}=0, {\hat{\chi }}+{\hat{\chi }}'=0 &{} {\hat{z}}=0,\\ {\hat{w}}\rightarrow 0, {\hat{\chi}} \rightarrow 0 &{} {\hat{z}}\rightarrow \infty , \end{array} \right\} \end{aligned}$$

(92)

has a non-trivial solution.

Fig. 16

Ground state solution for salt ${\hat{X}}^0$ using (38) (solid lines) and approximate solution ${\hat{U}}^0$ using (90) (dashed lines)

The equations in (92) can be combined into the fourth-order equation

$$\begin{aligned} L[w] := (D_{{\hat{z}}}^2+D_{{\hat{z}}}-{\hat{a}}^2)(D_{{\hat{z}}}^2-{\hat{a}}^2) {\hat{w}} = {\hat{a}}^2R^*e^{-{\hat{z}}}, \end{aligned}$$

(93)

where $D_{{\hat{z}}}$ denotes differentiation with respect to ${\hat{z}}$, resulting in the eigenvalue problem:

Given ${\hat{a}}>0$, find the smallest $R^*=R^*({\hat{a}})$ such that

$$\begin{aligned} \left. \begin{array}{lr} L[{\hat{w}}] = {\hat{a}}^2R^*e^{-{\hat{z}}} {\hat{w}} &{} {\hat{z}}>0,\\ \text {where } {\hat{w}} \text { fulfills } &{} \\ {\hat{w}}=0, (D_{{\hat{z}}}^2-{\hat{a}}^2)(D_{{\hat{z}}}+1){\hat{w}} =0 &{} {\hat{z}}=0,\\ {\hat{w}}\rightarrow 0 &{} {\hat{z}}\rightarrow \infty , \end{array} \right\} \end{aligned}$$

(94)

has a non-trivial solution.

Note that Eq. (93) also arises when studying the stability of the equilibrium state of the salt lake problem, see van Duijn et al. (2002) and references cited therein. In this study, the eigenvalue problem differs from (94) only through the second boundary condition of ${\hat{w}}$ at ${\hat{z}}=0$. In the salt lake problem, the second condition is ${\hat{\chi }}(0)=0$, implying that $D_{{\hat{z}}}^2{\hat{w}}(0)=0$.

Arguing as in van Duijn et al. (2002), we treat (94) by a semi-analytical technique based on a Frobenius expansion in terms of descending exponential functions:

$$\begin{aligned} {\hat{w}}({\hat{z}}) = \sum _{n=0}^\infty A_n(R^*)^ne^{(c-n){\hat{z}}}, \end{aligned}$$

(95)

where $c<0$. Substituting (95) into (93) gives the indicial equation

$$\begin{aligned} (c^2+c-{\hat{a}}^2)(c^2-{\hat{a}}^2)=0, \end{aligned}$$

yielding the negative roots

$$\begin{aligned} c_1=-{\hat{a}},\quad c_2 = -\frac{1}{2}-\sqrt{\frac{1}{4}+{\hat{a}}^2}, \end{aligned}$$

and the recurrence relation

$$\begin{aligned} \left\{ \begin{array}{lr} \frac{A_n^{(i)}}{A_{n-1}^{(i)}} = \frac{{\hat{a}}^2}{f_i(n;{\hat{a}})} &{} n\ge 1, \\ A_0^{(i)}=1, &{} \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} f_i(n;{\hat{a}}) = ((c_i-n)^2+(c_i-n)-{\hat{a}}^2)((c_i-n)^2-{\hat{a}}^2),\quad n\ge 1. \end{aligned}$$

Hence, we obtain the power series solution

$$\begin{aligned} {\hat{w}}({\hat{z}}) = A{\hat{w}}_1({\hat{z}}) + B{\hat{w}}_2({\hat{z}}), \end{aligned}$$

where

$$\begin{aligned} {\hat{w}}_i ({\hat{z}}) = \sum _{n=0}^\infty A_n^{(i)}(R^*)^ne^{-(c_i-n){\hat{z}}}, \quad i=1,2,\dots \end{aligned}$$

The boundary conditions at ${\hat{z}}=0$ are satisfied if

$$\begin{aligned} \big (\sum _{n=0}^\infty A_n^{(1)}(R^*)^n\big )A+\big (\sum _{n=0}^\infty A_n^{(2)}(R^*)^n\big )B=0 \end{aligned}$$

(96)

and

$$\begin{aligned} \big (\sum _{n=0}^\infty A_n^{(1)}g_1(n;{\hat{a}})(R^*)^n\big )A+\big (\sum _{n=0}^\infty A_n^{(2)}g_2(n,{\hat{a}})(R^*)^n\big )B=0, \end{aligned}$$

(97)

where

$$\begin{aligned} g_i(n;{\hat{a}}) = (c_i-n)((c_i-n)^2+(c_i-n)-{\hat{a}}^2),\quad n\ge 0. \end{aligned}$$

Writing (96) and (97) as

$$\begin{aligned} \begin{pmatrix} M \end{pmatrix}\begin{pmatrix} A\\ B \end{pmatrix}=\begin{pmatrix} 0\\ 0 \end{pmatrix}, \end{aligned}$$

we need to examine the characteristic equation $\det (M)=0$ to find the eigenvalues of (92). For $n\le 2$, we have

$$\begin{aligned} \det (M)&= \big (1+\frac{{\hat{a}}^2R^*}{f_1(1)}+\frac{({\hat{a}}^2R^*)^2}{f_1(2)f_1(1)}\big )\big (\frac{g_2(1)}{f_2(1)}{\hat{a}}^2R^* + \frac{g_2(2)}{f_2(2)f_1(1)}({\hat{a}}R^*)^2\big )\\&- \big (1+\frac{{\hat{a}}^2R^*}{f_2(1)} + \frac{({\hat{a}}^2R^*)^2}{f_2(2)f_2(1)}\big )\big ({\hat{a}}^2+\frac{g_1(1)}{f_1(1)}{\hat{a}}^2R^* + \frac{g_1(2)}{f_1(2)f_1(1)}({\hat{a}}^2R^*)^2\big ). \end{aligned}$$

For ${\hat{a}}\ll 1$, we take, to leading order, $n\le 1$ and obtain

$$\begin{aligned} \frac{g_2(1)}{f_2(1)}{\hat{a}}^2R^* = {\hat{a}}^2 + \frac{g_1(1)}{f_1(1)}{\hat{a}}^2R^*, \end{aligned}$$

or

$$\begin{aligned} R^*(\frac{g_2(1)}{f_2(1)}-\frac{g_1(1)}{f_1(1)}) = 1. \end{aligned}$$

Evaluating the coefficients yields

$$\begin{aligned} \frac{g_2(1)}{f_2(1)}-\frac{g_1(1)}{f_1(1)} = \frac{-4+O({\hat{a}}^2)}{8+O({\hat{a}}^2)} - \frac{-{\hat{a}}(1+{\hat{a}})}{{\hat{a}}(1+2{\hat{a}})} = -\frac{1}{2} + O({\hat{a}}^2) + \frac{1+{\hat{a}}}{1+2{\hat{a}}} = \frac{1}{2(1+2{\hat{a}})}+O({\hat{a}}^2). \end{aligned}$$

Hence,

$$\begin{aligned} R^*({\hat{a}}) = 2(1+2{\hat{a}}) + O({\hat{a}}^2) \text { as } {\hat{a}}\searrow 0. \end{aligned}$$

(98)

That means, in the limit as ${\hat{a}}$ approaches zero, we find

$$\begin{aligned} R^* = 2. \end{aligned}$$

Using again (91), we find that

$$\begin{aligned} R = \frac{2}{{\hat{X}}^0({\hat{t}},0)-1} \end{aligned}$$

(99)

approximates the critical Rayleigh number for ${\hat{a}}$ approaching zero. The Rayleigh numbers using the strategy from Sect. 3.5 with ${\hat{a}}=10^{-4}$ together with the approximate ones from (99) are shown in Fig. 17. The relative error between the approximate Rayleigh number and the Rayleigh number from Sect. 3.5 is larger for early times, as expected from Fig. 16 since the approximate ground state deviates more. However, the relative error is generally small, and decreases fast for later times. Hence, the approximate Rayleigh number (99) represents the behavior for low wavenumbers ${\hat{a}}$ well. We also see from (98) that $R^*$ (and hence R) decreases monotonously for low ${\hat{a}}$ as ${\hat{a}}$ approaches zero. This shows that ${\hat{a}}=0$ corresponds to a (local) minimum for $R({\hat{a}},{\hat{t}})$ for given ${\hat{t}}>0$.

Fig. 17

Critical Rayleigh number from Sect. 3.5 using ${\hat{a}}=10^{-4}$ and approximate Rayleigh number from (99), and the relative error between them as a function of non-dimensional time ${\hat{t}}$

E Grid and Time Step Convergence Study for the Numerical Simulations

Fig. 18

Influence of the grid cell size in x-direction on the onset time and the time of maximal mean value

Fig. 19

Influence of the height of the top cell $\varDelta z^\mathrm {top}$ on the onset time and the time of maximal mean value

A convergence study was performed to ensure that the spatial, and temporal discretization are able to capture the physical processes in sufficient detail and that the numerical diffusion has a negligible influence. The study was carried out using the same setup as described in Sect. 5 exemplary for the case with a fixed wavelength of $\lambda =0.01$ m and a permeability of $K=10^{-10}~\mathrm {m^2}$. As this case has the lowest used wavelength and hence the lowest resolution per perturbation, the influence of the discretization should be even less for longer wavelengths. The influence of the cell width in x-direction, the discretization in z-direction and the time step size was investigated. The onset time and the time of the maximal mean value $\mu _\mathrm {{\mathsf {x}}^\mathrm {NaCl}}^\mathrm {top}$ are used to evaluate the influence. A quadratic regression is used to extrapolate the values for a theoretical infinitesimal small cell width, cell height and time step.

The influence of the cell widths in x-direction on the onset time and time of the maximal mean value is shown in Fig. 18. The discretization in z-direction equals the one of the selected grid as described below. The investigated cell widths $5 \cdot 10^{-4}$ m, $1 \cdot 10^{-3}$ m, $2 \cdot 10^{-3}$ m and $3.33 \cdot 10^{-3}$ m correlate with 20, 10, 5 and 3 cells per initially applied perturbation wavelength. The selected grid uses 10 cells per perturbation wavelength. For longer wavelengths using lower permeabilities, this results in more cells per perturbation wavelength. The deviation of the selected grid from the extrapolated value is $6.3~\%$ for the onset time and $1.6~\%$ for the time of maximal mean value.

Figure 19 shows the investigation of the discretization in z-direction. In x-direction, the discretization equals the one of the selected grid as described above. In z-direction, a grading is used so that the top cell is the smallest, and the cell height increases toward the bottom of the domain. The same grading factor of 0.9 is used for the grids, but different numbers of cells. The different grids have 30, 35, 40 and 50 cells in z-direction, which correspond to a top cell height $\varDelta z^\mathrm {top}$ of $9.84 \cdot 10^{-4}$ m, $5.71 \cdot 10^{-4}$ m, $3.33 \cdot 10^{-4}$ m and $1.15 \cdot 10^{-4}$ m. Additionally, a grid with uniform grid heights of $1.0 \cdot 10^{-3}$ m was simulated. It has in total 200 cells in z-direction as it discretizes the lower parts of the domain finer, whereas $\varDelta z^\mathrm {top}$ nearly matches the coarsest graded grid. Nearly no influence could be observed from the finer discretization in the lower domain, on the time of onset and in maximal mean value. It is found that the height of the top cell has the main influence. We, therefore, look at the height of the top cell $\varDelta z^\mathrm {top}$ of the different grids. The selected grid uses grading and deviates $0.1~\%$ for the onset time and $0.2~\%$ for the time of maximal mean value from the respective theoretical extrapolated value. In comparison with the cell width, the cell height has less influence on the two evaluation times.

Further, an investigation of the time step size on the evaluation times was performed. It was found that for time steps from 25 to 150 s, the evaluation is independent of the time discretization. For all simulations, we use a time step of 50 s.