## 1. Introduction

Horizontal ice–water interfaces are ubiquitous in polar regions. They are found at the bottom of ice shelves, sea ice, and lake ice. In regards to ice shelves, bottom ablation in the Antarctic has caused an increasing glacier flow speed and glacier mass loss to the ocean over recent decades (Wouters et al. 2015). In regards to sea ice, bottom ablation in the Arctic has increased substantially in recent years (Perovich et al. 2013). The exact rates of bottom ablation cannot always be reconstructed from observational records and uncertainties remain of order one (Notz et al. 2003). In particular, it is not yet well understood how the turbulent motion of water beneath these interfaces influences the internal-energy flux that ablates the ice at the bottom. Here, we investigate the influence of turbulence on rates of bottom ablation of ice with laboratory experiments and direct numerical simulations.

The evolution of the flow in natural systems with horizontal ice–water interfaces is determined by several processes: buoyancy forcing, shear forcing, the effect of solutes, and the influence of surface roughness and heterogeneity. Rather than trying to capture the impact of all these processes on the interfacial internal-energy and salt fluxes, we here instead aim at simplifying the problem to its fundamental core. The simplified setup that we investigate is purely thermally driven free convection beneath a smooth fresh-ice–freshwater interface considering the density anomaly of the water.

However simple, our setup directly encompasses aspects of the flow underneath an ice–water interface from at least two more general systems: shear forcing and the effect of the solute salt. Shear forcing as well as buoyancy forcing create a similar vertical flow structure next to an interface (Pope 2000; Mellado 2012). Whenever turbulence aids the ablation of ice, a diffusive layer, however thin, forms between the ice and the turbulent layer irrespective of the forcing. A similar reasoning applies to the similarities between the present freshwater setup and a setup using seawater instead. With salt, the flow structure is qualitatively similar to the one induced by a thermally driven system. When ice is ablating on top of seawater, a lighter layer of almost freshwater resides aloft a denser layer of saline seawater. Nonlinear mixing of the layers can render a buoyancy reversal close to the interface. The resulting mean-buoyancy profile found beneath sea ice (Martin and Kauffman 1977) is similar to the mean-buoyancy profile that we focus on here, and we will discuss possible implications of our results of the fresh-ice–freshwater system for the more general case of ice floating on a salty ocean.

Martin and Kauffman (1977) examine free convection beneath sea ice in the laboratory. They study the temporal evolution of temperature and salinity profiles beneath floating ice. In their experiment, the ice floats on warm and salty water that is initially at rest. They maintain a constant far-field temperature and run the experiment until it reaches a steady state. From their bounded laboratory configuration and a one-dimensional theoretical model, they find an enhancement of the melt rate by a constant factor of about 2.5 compared to purely diffusive energy exchange. Their finding is based on the boundary layer structure they observe: a diffusive layer next to the interface overlies a layer of double-diffusive convection followed by thermal convection in the far field.

An unbounded, quasi-steady, free-convection system (the counterpart to a bounded, steady-state laboratory experiment) is hard to measure to satisfactory resolution in the field or laboratory. Direct numerical simulation offers an opportunity to study detailed boundary layer structures for free-convection systems that do not feel any bounding geometry and do not involve any turbulence model. Mellado (2012) provides the boundary layer structure of purely buoyancy-driven free convection over a heated plate. He forces free convection with a linear buoyancy function; thus, this work does not account for buoyancy reversal. Further studies do account for buoyancy reversal, such as in the cloud-top mixing layer (Siems et al. 1990; Wunsch 2003; Mellado 2010). Unfortunately, these studies do not account for the effect of a solid surface as intrinsic to convection beneath ice. In this paper, we extend this previous work to include the effect of a solid surface next to a region of buoyancy reversal, as occurs in convection beneath ice.

This study considers the melting of ice on top of thermally driven free-convective water with laboratory experiments and with direct numerical simulations. We employ the laboratory experiments to ensure that the simulations capture all relevant aspects of the physics. We employ direct numerical simulations to overcome resolution and size limitations of the laboratory experiment and yield statistically converged boundary layer mean profiles. From both laboratory experiment and simulations we obtain melt rates and boundary layer mean profiles. Based on the observed boundary layer structure, its temporal evolution, and the underlying dynamic, we derive an analytical expression for the melt rate that explains laboratory and simulation results. We then discuss our results in the context of more general systems, investigating in particular the impact of meltwater advection on the flow, the spatial inhomogeneity of the melt rate, and estimating the effect of salinity on the derived melt rates. We conclude that the turbulent enhancement of the melt rate of ice needs to be considered even in the absence of shear.

## 2. Setup and methods

A mass of solid, pure ice rests on top of an initially motionless body of pure water of fixed uniform temperature *T*_{∞} (Fig. 1a). The ice mass is isothermal at the freezing temperature of water *T*_{ice} and has a smooth surface. Hence, the ice imposes a Dirichlet boundary condition on the temperature field of the water and a no-slip boundary condition on the velocity field of the water. We consider the ice–water interface together with the water body as our system of interest.

*b*aswith Earth’s gravitational acceleration

*g*and water density

*ρ*(

*T*). Subsequently, we assume a second-order temperature dependence in the densitywith the temperature of maximum density

*T*

_{m}= 3.98°C,

*ρ*

_{m}= 999.96 kg m

^{−3}, and a thermal expansion coefficient

*β*= −7.00 × 10

^{−6}°C

^{−2}. This second-order approximation yields a relative deviation from Eq. (6) of Sharqawy et al. (2010) of less than 0.01% between 0° and 30°C.

With increasing distance from the interface, the water temperature increases from *T*_{ice} at the interface to *T*_{∞} in the far field (Fig. 1b). As the water temperature increases, the density first increases (as long as *T* < *T*_{m}) and then decreases (when *T* > *T*_{m}; Fig. 1c). Accordingly, the buoyancy first decreases and then increases again. Hence, for *T*_{∞} > *T*_{m}, the buoyancy in the far field is higher than the buoyancy closer to the interface, and the system is convectively unstable (Fig. 1d). We are interested in this convectively unstable regime.

The temperature difference Δ*T*_{∞} = *T*_{∞} − *T*_{ice} divides the system into two convective regimes based on Δ*T*_{m} = *T*_{m} − *T*_{ice}. For Δ*T*_{∞} > 2Δ*T*_{m}, the whole column of fluid can overturn (Fig. 1d, reddish profiles). In contrast, for Δ*T*_{∞} < 2Δ*T*_{m}, there exists a layer next to the interface that is stably stratified (Fig. 1d, bluish profiles). This stably stratified layer beneath the ice acts as a shield against the warmer fluid of the convectively unstable region, as we will show below.

**v**(

**x**,

*t*), temperature field

*T*(

**x**,

*t*), spatial coordinate

**x**=

*x*

_{1}

**e**

_{1}+

*x*

_{2}

**e**

_{2}+

*x*

_{3}

**e**

_{3},

**e**

_{i}=

*ε*

_{ijk}

**e**

_{j}

**e**

_{k}, and time

*t*, follows the evolution equations for mass, momentum, and internal energy:We have given the equations in the Boussinesq approximation. The variable

*ν*is the kinematic viscosity,

*κ*is the thermal diffusivity,

*p*is the modified kinematic pressure, ∂

_{t}is the temporal derivative, and ∂

_{i}is the spatial derivative in direction

**e**

_{i}.

*b*[Eq. (3b)]. For the analysis of the problem, it proves useful to express the dependence of the buoyancy on the temperature, given by Eq. (1), asThe minimum buoyancy

*b*

_{m}is then given byThe variable

*θ*is a normalized temperaturewhich varies between 0 in the far field and 1 at the interface. For convectively unstable conditions,

*T*

_{∞}>

*T*

_{m}, the far-field temperature parametervaries between 0 and 1. The far-field temperature parameter

*θ*

_{m}denotes the fraction of the temperature range [

*T*

_{ice},

*T*

_{∞}] that locally stratifies the fluid unstably, and (1 −

*θ*

_{m}) denotes the fraction that locally stratifies the fluid stably.

The flow develops freely into the far field and does not feel any solid boundary but the ice–water interface. For sufficiently low viscosity, the system becomes turbulent, decorrelates from its initial state after a sufficiently long time, and depends solely on the set of control parameters {*ν*, *κ*, *b*_{m}, *θ*_{m}}. Dimensional analysis provides the set of independent control parameters {Pr, *θ*_{m}}, with Pr = *ν*/*κ*. Here, we chose to nondimensionalize the system by *b*_{m} instead of by the full buoyancy difference across the system because *b*_{m} gives the convective instability driving the flow. We constrain this investigation to waterlike fluids of the fixed Prandtl number Pr = 10. Hence, any flow property solely depends on the governing parameter *θ*_{m} and on the position in space and time {*θ*_{m}, **x**, *t*}. The fully developed turbulent system is statistically homogeneous in horizontal directions. We denote horizontally averaged quantities by 〈⋅〉 and fluctuations around that mean by a prime. Horizontally averaged statistics only depend on {*θ*_{m}, *z*, *t*} with *z* = −**xe**_{3} and the origin of **x** chosen such that *z* gives the distance from the interface. The terms *θ*_{m}, *b*_{m}, and *T*_{∞} equivalently describe the system [Eqs. (5) and (7)].

### a. Laboratory experiment

The laboratory setup that mimics the system consists of a Plexiglas tank filled with tap water, a thermistor cascade just beneath the water surface, and an ice block dangling on a manual tackle just above the water surface. The setup resides in a cold room whose temperature can be controlled with a precision of ±3°C. The water body has the dimensions (0.347 m)^{3} and is laterally isolated by a Styrofoam cover around the tank. The height of the water tank is *h*_{water} = 0.347 m. Temperature changes are measured with a cascade of 23 thermistors. The cascade has a spacing of 7 mm between the thermistors and a total profile length of 154 mm. Each thermistor is spherical with a radius of 1 mm and is placed on the tip of a 4-mm-thick and 2-cm-long finger to minimize the influence of the instrument body on the flow. The precision of the temperature measurement is at least 0.01 K, while temperature changes are measured with a response time of 2 to 4 s depending on the sign of the temperature change. The ice block, 8- to 13-cm thick, covers almost the entire water surface and is prepared from distilled water prior to the experiment.

Before an experiment is conducted, we orientate the system components and prepare the temperatures of the water body and the cold room. We level the ice block such that its surface can be attached evenly to the water surface, and we displace the thermistor cascade such that the upper thermistor is just beneath the water surface. A pump mixes the water body to a homogenous initial temperature *T*_{∞}. The cold-room temperature matches *T*_{∞} to minimize any temperature gradient between the water body and its lateral surrounding. We keep the cold-room temperature constant at *T*_{∞} for several hours to ensure that the ice is isothermal. The melting of the ice due to its exposure to the cold-room air is little compared to the melting due to its exposure to the water body later on. When the pump stops, we await the decay of turbulence kinetic energy for 3 min and cautiously lower the ice block via the manual tackle.

The experiment starts as soon as the ice touches the water surface. The ice is not lowered further. The thermistor cascade measures a temperature profile every 5 s during the experiment run time *t*_{run}. After that, the ice block is lifted from the water surface and the recording continues for a short while as the water is mixed with the pump to record the final mean temperature of the system *T*_{end}. This entire procedure constitutes one realization of the laboratory experiment. The horizontal position of the thermistor cascade is different in each laboratory realization.

First, we focus on the setup of *T*_{∞} = 5°C because considerably higher and lower far-field temperatures handicap the measurement. For higher far-field temperatures, the system evolves too quickly compared to the response time of the thermistors. For lower far-field temperatures, the expected temperature change in the far field is too little compared to the signal-to-noise ratio. We record an ensemble of 25 realizations at *T*_{∞} = 5°C and *t*_{run} = 15 min to capture the mean-temperature profile of the turbulent system. Second, we record several ensembles of small realization number (three to five) at *T*_{∞} ∈ {4.5, 5, 6, 14.8}°C and varying *t*_{run} to capture the temporal bulk temperature change of the turbulent system.

### b. Direct numerical simulation

We integrate Eqs. (3) using a high-order, finite-difference method on a collocated, structured grid. We approximate the integration by a fourth-order Runge–Kutta scheme and the spatial derivatives by sixth-order, spectral-like finite differences (Williamson 1980; Lele 1992). After every integration step, a pressure solver ensures fulfillment of the solenoidal constraint. For this we use a Fourier decomposition along periodic horizontal coordinates and a factorization of the resulting second-order equations in the vertical coordinate (Mellado and Ansorge 2012).

The calculations are performed on a grid of 576 grid points in the vertical direction and 1280 grid points in both horizontal directions. Adequacy of vertical resolution, domain height, and domain width has been assured so that the results discussed in this paper are sufficiently independent of those simulation properties. The grid spacing is uniform in the horizontal directions and in most of the vertical direction. The resolution in the vertical direction close to the interface, however, is increased because the main mean-temperature variation all over the domain occurs close to the interface. This temperature variation potentially entails the main mean-buoyancy change, a change in the forcing of the system from a positive to the global-extreme negative value and back to almost zero (Fig. 1d). To fully cover this buoyancy variation, we increase the resolution next to the interface by a factor of 5. The regions of uniform and adjusted resolution along the vertical direction are gradually matched by hyperbolic tangents. Finally, the grid in the vertical direction far from the interface is coarsened to save computing time. This part of the domain serves to diminish the influence of the computational boundary on the flow.

The boundary conditions in the velocity field are no-slip and no-penetration at the interface and free-slip and no-penetration in the far field. The boundary conditions in the temperature field are Dirichlet at the interface and Neumann in the far field. The initial condition in the temperature field is an error function of thickness *d* and zero in the velocity field. The temperature field is perturbed close to the interface with a wavelength that corresponds to 7*d* [see Mellado (2012) for details].

We simulate systems of different far-field temperatures *T*_{∞} (Table 1). We study the system of Δ*T*_{∞} = 2Δ*T*_{m} (*θ*_{m} = 0.5) and three systems of both higher and lower buoyancy at the interface. The simulation of *θ*_{m} = 0.179 is done on an extended grid of size 2560 × 1088 × 2560 to reach higher Reynolds numbers, to increase the statistical convergence, and to calculate the melt rate more accurately. In addition, we perform three numerical simulations with altered flow boundary conditions to study the influence of meltwater on the flow.

Properties of the numerical simulations. Far-field temperature parameter *θ*_{m}, far-field temperature *T*_{∞}, magnitude of minimum buoyancy |*b*_{m}|, and Richardson number Ri_{0} equivalently define the simulation [Eqs. (5), (7), and (26)] and are given for the reader’s convenience. The diffusive length scale *z*_{0} is defined in Eq. (15). The boundary layer height *z*_{*} and the convective velocity scale *w*_{*} are defined in Eqs. (16) and (17), respectively. The turbulent Reynolds number Re_{turb} is the maximum value of *e*^{2}/(*εν*) in the domain with turbulence kinetic energy *e*, viscous dissipation rate *ε*, and viscosity *ν*. The terms Re_{turb} and *w*_{*}*z*_{*}*ν*^{−1} are measures for the scale separation in the simulations; *η* is the domainwide minimum Kolmogorov scale. The diffusive velocity scale is *w*_{0} = (*z*_{0}*b*_{m})^{1/2}. The melt rate is given with Eq. (21). Columns 6–10 are evaluated at the final time step of the simulations. The grid size of the simulations is 1280 × 576 × 1280.

The final boundary layer height of the simulated systems *z*_{*} varies between 0.12 and 1.05 m. The simulations reach Reynolds numbers *w*_{*}*z*_{*}/*ν* and *e*^{2}/(*εν*) of up to 3600 and 1000, respectively, with the turbulence kinetic energy *e*, the viscous dissipation rate *ε*, the convective velocity scale *w*_{*}, and the viscosity *ν*.

## 3. Similarity of laboratory experiment and numerical simulation

We compare an ensemble of 25 laboratory realizations of *T*_{∞} = (5.00 ± 0.01)°C to one simulation of *T*_{∞} = 4.98°C (*θ*_{m} = 0.20). The simulation grid needed to resemble a laboratory tank of domain size 0.360 m is just 1024 × 384 × 1024 grid points, where we choose an aspect ratio of eight to one instead of one to one. Within a horizontal cross section of 1024 × 1024 grid points, we choose 256 equally distributed base locations to obtain tower data measurements. The tower data consist of the temporal evolution of the vertical temperature profile beneath the ice interface, just as a laboratory realization does. We qualitatively compare the phenomenology obtained from simulation visualizations and laboratory realizations with dyed ice, and we quantitatively compare the temporal evolution of both the mean behavior and individual tower data of the simulation and the laboratory experiment.

Qualitatively, the structure of the boundary layer beneath the ice in the laboratory matches the one in the simulation (Fig. 2). Cold fluid (bright) unites more mass per volume and congregate into thin streaks that push downward through warmer surrounding fluid (dark). The lowermost point of the streaks is marked by a plumelike tip structure with vortices at their sides. Looking at the horizontal cross section, a honeycomb pattern is seen for both laboratory experiment and numerical simulation. From the comparison of vertical and horizontal cross sections, one observes that plumes actually form the septum of the honeycomb cells. Cold plumes push down along the cell rim while fluid must consequently move up in the interior of the cell. Plumes are the dominant structures. Their movement and diffusive decay mixes the water beneath the ice in the so-called mixed layer. The mixing successively entrains calm, warmer water from below and the colder mixed layer broadens. Both laboratory experiments and simulations exhibit this working principle, known from the convective boundary layer (CBL) in the atmosphere (Stull 1988), but vertically inverted, and from Rayleigh–Bénard convection (Chillá and Schumacher 2012).

Quantitatively, the time evolution of temperature profiles of individual laboratory realizations, as well as that of individual tower data from the simulation, varies strongly among each other. Hence, we draw the comparison on the basis of the mean-temperature profiles instead of on the basis of individual laboratory realizations and individual tower data from the simulation.

The mean profiles evolve as described in the previous paragraph: a mixed layer develops and broadens in time. The broadening happens at the same pace in the laboratory experiment as in the simulation (not shown) and takes about 7 min to cover a height of 0.40*h*_{water}, with the tank height *h*_{water} = 0.347 m. In the mixed layer, we find a mean temperature almost constant in time. At the interface, the mean normalized temperature decreases by an order of magnitude over a distance of ~0.05*h*_{water} for both laboratory and simulation. This region forms part of what is generally referred to as “the surface layer” (Stull 1988).

Differences between the laboratory experiment and simulation exist in the boundary conditions, the initial conditions, and the system itself. Moreover the presence of the measuring device alters the flow. We hence expect some differences between the laboratory experiment and the numerical simulations but find that such differences are statistically insignificant.

To show this, we perform a Kolmogorov–Smirnov test on the mean profiles with the following null hypothesis: the laboratory measurements and simulation tower data are drawn from the same distribution. We cannot reject the null hypothesis on a significance level of 5% for the regions indicated by the gray bars in Fig. 3a. The laboratory measurement and tower data are thus to be considered as statistically equivalent in those regions. The regions broaden with evolving time and stretch from close to the interface to roughly half of the mixed layer.

In summary, we find qualitative and quantitative agreement between the simulation and laboratory experiment for the given number of laboratory realizations. Both experiments and simulations are used in the following section to investigate the melt rate of the ice.

## 4. The melt rate

*H*

_{water}is supplied to the ice interface than can be transported away (

*H*

_{ice}). We focus on an isothermal ice block with

*H*

_{ice}= 0. Hence, the melt rate

*w*

_{f}is solely determined by

*H*

_{water}, according towhere

*L*= 333.5 kJ kg

^{−1}is the specific energy required to melt the ice (Frankenstein and Garner 1967), and

*ρ*

_{ice}= 916.8 kg m

^{−3}is the density of pure ice at 0°C (Pounder 1965). The internal-energy flux at the ice interface

*H*

_{water}originates from the evolution of the internal energy in the fluid system. The mean evolution of the internal energy [Eq. (3c)] iswithThe specific heat capacity of water is

*c*

_{p}= 4.22 kJ kg

^{−1}K

^{−1}; the thermal diffusivity of water is

*κ*= 1.36 × 10

^{−7}m

^{2}s

^{−1}at 273.15 K (Sharqawy et al. 2010) and

*ρ*

_{water}=

*ρ*(

*T*

_{∞}).

### a. Melt rates observed in the laboratory experiment

*E*

_{int}=

*m*

_{water}

*c*

_{p}(

*T*

_{end}−

*T*

_{∞}) over the experiment run time

*t*

_{run}. After the run time

*t*

_{run}, the initial mean bulk temperature

*T*

_{∞}has decreased to its final value

*T*

_{end}. The net change of the internal energy Δ

*E*

_{int}is converted to the heat of fusion

*m*

_{ice}

*L*and to internal energy of the meltwater

*m*

_{ice}

*c*

_{p}(

*T*

_{end}−

*T*

_{ice}). From the integral formulation of Eq. (9) one obtains the internal-energy balance of the system:where

*m*

_{ice}is the mass of the melted ice, and

*m*

_{water}is the mass of the water in the tank.

Equation (11) assumes that no energy flows through water body boundaries except through the ice–water interface. To test this assumption, we estimate the mean energy flow through water-body boundaries other than the ice–water interface from time series of the bulk water temperature without ice attached on the water. We find a relative uncertainty in Δ*E*_{int} due to all neglected mean energy fluxes of about 15%.

*A*, from

*m*

_{ice}=

*ρ*

_{ice}

*h*

_{ice}

*A*and from

*m*

_{water}=

*ρ*

_{water}

*h*

_{water}

*A*, the height

*h*

_{ice}of the melted ice is given asSeveral laboratory experiments for the same temperature

*T*

_{∞}, but with different run time

*t*

_{run}, allow us to estimate a melt rate

*w*

_{f}= Δ

*h*

_{ice}/Δ

*t*

_{run}from the data. This assumes a constant energy flux at the interface, which is the leading-order behavior for free-convection systems and which we observe for all our simulations (presented in section 4c). The standard deviation among the calculated heights at each

*t*

_{run}is less than 13%. We obtain the melt rates

*w*

_{f}= [53, 55, 136, 1260] mm day

^{−1}for

*T*

_{∞}= [4.5, 5, 6, 14.8]°C within the error bounds supplied in Fig. 4 (crosses and black solid bars). We compare these melt rates

*w*

_{f}to the melt rates

*w*

_{d}of a purely diffusive experiment that has a similar boundary layer extent. To allow for such comparison, we need an expression that describes how the melt

*w*

_{d}depends on boundary layer extent, which can be obtained from the derivative of a diffusive temperature profile

*T*

_{d}(

*z*):where

*T*

_{d}(

*z*) is assumed to be an error function, a solution of the diffusion equation for the Dirichlet boundary conditions

*T*

_{d}(

*z*= 0) =

*T*

_{ice}and

*T*

_{d}(

*z*→ ∞) =

*T*

_{∞}. The boundary layer height

*δ*is defined based on the threshold criterion [〈

*T*

_{d}(

*z*=

*δ*)〉 −

*T*

_{ice}]/Δ

*T*

_{∞}= 0.995. In the following, we choose

*δ*=

*h*

_{water}, so that

*w*

_{d}represents melt rates of diffusive experiments of the size of the laboratory tank.

We find that for the same boundary layer extent, the melt rates *w*_{f} are increased by one to two orders of magnitude compared to *w*_{d} (Figs. 4b,c), in contrast to the increase by a constant factor of about 2.5 reported in Martin and Kauffman (1977). We will elaborate on this finding in section 5a.

### b. Melt rates simulated in the numerical experiment

*H*

_{water}according to Eqs. (8) and (10). At the interface (

*z*= 0) one finds

We obtain melt rates for several simulations of varying far-field temperature (Table 1, Fig. 4a, dots). We can infer two important results: First, the melt rates of the simulations and of the laboratory experiments match within the limits of their respective uncertainties. [The relative error of the melt rates from the simulation is less than 10% due to varying initial conditions (Mellado 2012).] Second, the melt rates depend strongly on the far-field temperature. From temperatures *T*_{∞} = 3.98°C to *T*_{∞} = 5.20°C, melt rates increase strongly from diffusive values of 11 mm day^{−1} (dashed line) up to 86 mm d^{−1} (Fig. 4b, blue dot), that is, a factor of ≈8. From temperatures of *T*_{∞} = 17°C to *T*_{∞} = 40°C, melt rates increase from 1.7 to 7 m day^{−1} (Fig. 4c, gray dot and plus), that is, a factor of ≈4.

The simulation of *θ*_{m} = 1 cannot be assigned to a definite far-field temperature; *θ*_{m} = 1 is the limit of high far-field temperatures [Eq. (7)] and of negligible buoyancy reversal. For this simulation, we find the minimum buoyancy directly at the wall (Fig. 1d, dashed line). This situation is reminiscent of the free convection over a heated plate (Mellado 2012), where buoyancy increases linearly with the stratifying agent from the minimum value at the wall to the far field. We provide the melt rate derived from our simulation of *θ*_{m} = 1 exemplarily for a far-field temperature of *T*_{∞} = 40°C, for which we expect the assumption of negligible buoyancy reversal to be a sufficiently good approximation (Fig. 4, plus).

## 5. The energy flux and the flow structure

The partitioning of the internal-energy flux according to Eq. (10) into a molecular flux −*c*_{p}*ρκ*∂_{3}〈*T*〉 and a turbulent flux *H*_{water}. The constant flux foretells, according to Eq. (9), a steady mean-temperature profile despite cooling from the interface. A large extent of the system is hence in dynamic equilibrium (Mellado 2012). As a consequence, we expect a steady temperature gradient at the interface, that is, a constant melt rate.

The illustrated working principle and the internal-energy flux partition warrant a separate description of the diffusion-dominated inner layer and the turbulence-dominated mixed layer.

### a. The diffusion-dominated inner layer

*z*

_{0}. The layer is stably stratified except for an unstably stratified fraction at its lower bound. This unstable fraction tends to overturn and hence tends to thin the diffusion-dominated layer. Such a situation is reminiscent of the buoyancy-reversal configuration studied, for example, by Siems et al. (1990) and Mellado (2010). The overturning fraction covers the normalized-temperature range [0, 2

*θ*

_{m}] [Eq. (4) and Fig. 1d] and thus extends over a distance 2

*θ*

_{m}

*z*

_{0}. A buoyancy perturbation within 2

*θ*

_{m}

*z*

_{0}accelerates the flow proportionally to

*b*

_{m}, whereas the viscosity decelerates the flow proportionally to

*ν*. The time scale of the perturbation growth is

*t*

_{buoy}= (

*ν*/

*b*

_{m})/(2

*θ*

_{m}

*z*

_{0}) (Turner 1973). On the other hand, diffusion broadens

*z*

_{0}at a rate

*κ*/

*z*

_{0}. The time scale of diffusive advancement over the overturning fraction 2

*θ*

_{m}

*z*

_{0}is hence

*t*

_{diff}= (2

*θ*

_{m}

*z*

_{0})/(

*κ*/

*z*

_{0}). While the diffusive advancement increases

*z*

_{0}, the buoyancy perturbation accelerates fluid away from the region and decreases

*z*

_{0}. The critical depth at which the rate of thinning is equal to the rate of broadening,

*t*

_{diff}=

*t*

_{buoy}, marks the extent of the diffusion-dominated layer

*z*

_{0}:The layer of depth

*z*

_{0}is “the diffusive sublayer,” a part of the surface layer that has been introduced in the previous section. The factor 10 gives the commonly used criticality for this Rayleigh number criterion. The depth

*z*

_{0}is independent of time; the diffusive sublayer has a fixed extent.

### b. The turbulence-dominated mixed layer

*ε*balances a large constant fraction of the turbulent buoyancy production

### c. The energy flux of fully developed systems

^{2}K m

^{−1}at

*T*

_{∞}≈ 4.8°C to 3.7 × 10

^{3}K m

^{−1}at

*T*

_{∞}= 9.6°C. Much of this dependence on

*T*

_{∞}can be explained from the previous analysis of the diffusion-dominated inner layer. From this analysis, we know that this gradient is described to first order by a temperature drop of Δ

*T*

_{∞}over a length

*z*

_{0}:with the proportionality constant

*f*only dependent on the far-field temperature parameter

*θ*

_{m}. Figure 6b confirms this estimate, although

*f*still varies within the interval [1,0; 2.2].

*z*

_{0}has been derived as the critical depth at which molecular diffusion and buoyancy are in balance. This reasoning was inspired by the analogy between this study and the cloud-top mixing layer. The system in this study, however, has a solid wall next to the diffusive sublayer. Hence, we attribute the variation

*f*of the normalized temperature gradient at the wall (Fig. 6) to the presence of the wall. In the absence of a theoretical description of the influence of the wall, we suggest accounting for it by an empirical first guess:(Fig. 6, dotted line).

## 6. The melt rate as a function of the far-field temperature

*T*

_{∞}> Δ

*T*

_{m}. Substituting the definition of

*z*

_{0}[Eq. (15)] into the flux parameterization [Eq. (19)], the melt rate, given by Eqs. (8) and (14), can be written aswhere Eqs. (5), (7), and (20) have been used to express the dependency in terms of

*T*

_{∞}instead of

*θ*

_{m}. The prefactor iswith Pr = 10 (Fig. 4, black solid line). The term

*w*

_{f,0}corresponds to 1.86 × 10

^{2}mm day

^{−1}. Note that in nature Pr = Pr(

*T*) and varies from 13.18 to 5.40 in the relevant temperature range of 0° to 30°C (Sharqawy et al. 2010). The melt rates of Eq. (21) match the melt rates estimated in the laboratory experiment and in the simulations. Equation (21) seems to slightly underestimate the simulation results for low far-field temperatures (Fig. 4b), but not all these simulations have yet completely reached their steady behavior.

The enhancement of the melt rate *w*_{f}/*w*_{d} increases by a factor of 40 as the far-field temperature increases from 4° and 8°C (Fig. 4b). The enhancement of the melt rate increases by a factor of about 1.5 as the far-field temperature increases from 20° to 40°C (Fig. 4c). Thus, the relative enhancement of the melt rates with far-field temperature decreases for increasing temperature.

*z*

_{diff}= (

*κ*

^{2}/|

*b*

_{m}|)

^{1/3}, one finds

In the high temperature limit, (*T*_{∞} − *T*_{m})/Δ*T*_{m} ≫ 1, we obtain (Δ*T*_{∞}/Δ*T*_{m})^{5/3} from Eq. (23), and we also obtain (Δ*T*_{∞}/Δ*T*_{m})^{5/3} from Eq. (21) (Fig. 4, dashed line). Thus, the high temperature limit of Eq. (21) corresponds to the behavior found for the heated plate, where no stable stratification is present. We conclude that the shielding of the interface by the stable stratification next to it effectively vanishes for *T*_{∞} ≫ *T*_{m}.

In the low temperature limit, (*T*_{∞} − *T*_{m})/Δ*T*_{m} < 1, we obtain [(*T*_{∞} − *T*_{m})/Δ*T*_{m}]^{2/3} from Eq. (23) in contrast to [(*T*_{∞} − *T*_{m})/Δ*T*_{m}]^{4/3} from Eq. (21). Thus, the shielding of the interface—incorporated in Eq. (21) but not in Eq. (23)—diminishes the melt rate by a power of two if it is expressed as a function of the relative temperature difference (Δ*T*_{∞} − Δ*T*_{m})/Δ*T*_{m} < 1.

## 7. Discussion

### a. The influence of meltwater on the melt rate

When ice melts, it forms meltwater of *T* = 0°C. Recently formed meltwater does not possess any kinetic energy and thickens the diffusive sublayer in which the stable stratification occurs. Thus, meltwater increases the shielding of the ice from rising warm fluid and diminishes the melt rate. This describes a negative feedback. We now assess its strength.

*w*

_{ice}, the formation of meltwater appears as vertical background mean advection. The corresponding advection velocity isIn principle,

*w*

_{ice}depends on time and on the position within the interface. We have seen already that the time dependence of the mean value vanishes after an initial transient, when the system is freely developing (Fig. 6). For now, we also assume no dependence on the position within the interface, but we will assess this assumption in the next section.

With constant *υ*_{int}, the moving frame of reference is still an inertial system. The governing Eqs. (3) still apply, but different boundary conditions need to be considered. Instead of no-slip, no penetration boundary conditions, the system imposes no-slip, constant velocity boundary conditions *υ*_{i}(*z* = 0) = *υ*_{int}*δ*_{i3}.

*w*

_{ice}=

*w*

_{f}in Eq. (24):For far-field temperatures approaching

*T*

_{m}, the estimate approaches (

*c*

_{p}/

*L*)(

*T*

_{ice}−

*T*

_{m}) ≈ 1/20 and diverges for high far-field temperatures (Fig. 7, line). The divergence for high far-field temperatures exposes that the estimate of Eq. (25) does not account for the full feedback mechanism described above. Meltwater cannot diminish the melt rate infinitely but only diminishes it as long as ice is melting. Here, we only considered a fixed constant velocity boundary condition with

*w*

_{ice}determined from Eq. (21). So Eq. (25) is merely a maximum estimator to the feedback mechanism, and we expect to find a smaller diminution in practice than estimated here.

To verify these estimates, we conduct a set of three simulations at temperatures *T*_{∞} ∈ {8, 25, 40}°C that account for the meltwater formation with *υ*_{int} set accordingly. We compare the melt rates of these simulations *w*_{f} obtained from Eq. (21) as *w*_{f}(*T*_{∞}) (Fig. 7, dashed line). The simulated diminution and Eq. (25) follow a similar dependence on *T*_{∞}, with an offset of ≃−8% and a shift of ≃0.06*θ*_{m}. Hence, the velocity *υ*_{int} of each of the three simulations was systematically chosen too high to correctly represent reality. Smaller velocities *υ*_{int} would in fact diminish the melt rate less and yield melt rates closer to Eq. (21).

We find the numerical estimate in agreement with the laboratory experiments. We did not observe a pronounced signature of the meltwater in the flow structure as compared to the simulations with no penetration boundary conditions.

We conclude that the effect of meltwater formation can at least be neglected for far-field temperatures *T*_{∞} < 20°C (*θ*_{m} = 0.8). Within this limit the presented melt-rate equation [Eq. (21)] holds with a diminution of less than 10% due to the influence of meltwater (Fig. 7, black dashed lines). For far-field temperatures *T*_{∞} ≫ 20°C, for which the influence of the meltwater is significant, we expect the feedback to yield a temperature dependence in between *w*_{f} ∝ Δ*T*_{∞} (diffusive) and

### b. Spatial inhomogeneity of the melt rate

The temperature gradient at the interface (Figs. 8a,b) varies in space (*x*_{1}, *x*_{2}). We now assess this variation of the melt rate around its mean value [Eq. (21)] on the basis of the probability density function (pdf; Fig. 8c).

The ensemble of pdfs for different far-field temperatures exhibits two main modes of different characteristic. The first mode is dominant in pdfs of low far-field temperatures, represents melt rates smaller than the mean, and has a relative standard deviation of 10%. The second mode is dominant in pdfs of high far-field temperatures, represents the mean, and has a relative standard deviation of 60%. In the diffusive limit, that is in the absence of convection, one expects to find a pdf that is a delta function at 〈∂_{3}*T*〉|_{z=0}. The first mode is considerably narrower than the second mode and yields absolute melt rates close to those of purely diffusive melting (Fig. 4). We refer to the first mode as the diffusive mode. The second mode at the center of the pdf resembles the melt rate in the high temperature limit *θ*_{m} = 1 (Fig. 1d, dashed curve; cf. Fig. 8c, black). This mode is purely convection dominated, as can be inferred from the interface pattern (Fig. 8b). We refer to the second mode as the turbulent mode.

With increasing far-field temperature, we observe a shift of the main contribution to the pdf from the diffusive mode to the turbulent mode. This transition is in line with the mechanism proposed so far; for low far-field temperatures, a strong stable stratification close to the interface shields the ice from the rising warm turbulent water. Only a few rising plumes manage to intrude this shield and enhance the melting at the interface locally (Fig. 8a). This local turbulent intrusion contributes to the pdf at values of up to several times larger than the first mode. For high far-field temperatures, the strength of the stable stratification is not sufficient to shield the ice anymore. The increasing contribution of the turbulent mode broadens the absolute bandwidth of the pdf (not shown) and shifts the diffusive mode to lower relative values. As the importance of the diffusive mode ceases, the relative bandwidth of the pdf reduces. Then, the ice is immediately exposed to rising, warm turbulent plumes, and we find an interface pattern known from the turbulent free-convective motion (Fig. 8b).

We found in the last section that the influence of the meltwater formation is negligible for low far-field temperatures. For increasing far-field temperatures, the shielding effect of the meltwater becomes more relevant, according to Eq. (25). Based on this result, we can infer that the maintenance of a spatially resolved *υ*_{int}(*x*_{1}, *x*_{2}) in section 7a would diminish the high temperature end of the pdf more than it would diminish the low temperature end of the pdf. This would homogenize the interface pattern found in Fig. 8b and would narrow the bandwidth of the probability density function in Fig. 8c.

### c. Application of this work to the ice–ocean interface

The motivation to study the present setup has been its structural similarity to the sea-ice–ocean interface and to the ice-shelf–ocean interface. The question arises of how much information can be transferred from our findings to such interfaces. We are aware that ice–ocean interfaces do not just melt but ablate in the interplay of differently diffusing salinity and temperature. Still, we use our model as a simplified approach and focus only on the shape of the mean-buoyancy profile.

*b*(

*z*= 0) >

*b*

_{m}. We define the relative strength of the shielding as the strength of the stable stratification Δ

*b*=

*b*(

*z*= 0) −

*b*

_{m}compared to the strength of the buoyancy forcing

*b*

_{m}(Fig. 1d). This is effectively the Richardson numberthat describes the ratio between the potential energy that a fluid particle requires in order to overcome the diffusive shield Δ

*b*of thickness

*z*

_{0}and the kinetic energy

*b*

_{m}over a distance

*z*

_{0}.

*b*(

*z*= 0) ≈ 0.1385 m s

^{−2}from their salinity and temperature profiles given in Fig. 2e and Eq. (9) of their study together with Eq. (1) of this study. (The zero-order density is taken as 10

^{3}kg m

^{−3}and not as 10

^{2}kg m

^{−3}as given in their study.) With a redefinition of the minimum buoyancy that accounts for the influence of temperature and salinity,using the far-field salinity

*S*

_{∞}taken from their salinity profile, we find

*b*

_{m}≈ 0.0011 m s

^{−2}and Ri

_{0}≈ 129. If we use the boundary conditions

*T*

_{ice},

*S*

_{∞}, and

*T*

_{∞}as provided from their one-dimensional theoretical model, we consistently obtain Ri

_{0}≈ 136 for their setup. In the case studied in this work, we find a similarly strong stable stratification for a far-field temperature of

*T*

_{∞}= 4.33°C.

According to our results, turbulence enhances the melt rate by a factor *w*_{f}(*T*_{∞} = 4.33°C)/*w*_{d}(*T*_{∞} = 4.33°C) ≈ 3.1. Martin and Kauffman (1977) determined an enhancement of the melt rate by a constant factor of about 2.5. Given that we consider melt-rate variations over two orders of magnitude (Fig. 9a), this is in fairly good agreement despite the neglect of the influence of salt.

As opposed to the finding of Martin and Kauffman (1977), we find that turbulence does not enhance the melt rate by a constant factor, but by a factor that depends on the relative strength of the shielding, the Richardson number Ri_{0}. With the parameter range of far-field temperature and salinity that they provide from their model (Fig. 9, therein), Ri_{0} varies between 10^{2} and 10^{3} in their study. This order-of-magnitude variation in Ri_{0} translates to a variation in the turbulent enhancement *w*_{f}/*w*_{d} by a factor of approximately 4 according to Fig. 9a (herein). For *T*_{∞} ∈ [10, 30] °C, Ri_{0} even yields *O*(10^{1}); the turbulent enhancement of the melt rate for Ri_{0} = *O*(10^{1}) can be twice that of Ri_{0} = 136. We conjecture that Martin and Kauffman (1977) would have found a turbulent enhancement of the melt rate by a factor of 5 for significantly increased far-field temperatures.

*T*

_{∞}as the independent variable. The direct link of the buoyancy-forcing strength

*b*

_{m}to the velocity scale of the system

*w*

_{0}is replaced by the more generic convective velocity scale of the system

*w*

_{*}, as defined in Eq. (17). The effect of

*T*

_{∞}is still retained in Δ

*b*. This Richardson number describes the importance of the shielding by the diffusion-dominated, inner-layer relative to the specific kinetic energy contained in the turbulence-dominated mixed layer (Fernando and Hunt 1997). One could thus interpret the simulations of different far-field temperatures as simulations of fixed far-field temperature but different Ri

_{*}(Fig. 9b). From a linear fit of the final simulation values log(

*w*

_{f}/

*w*

_{d}) over log(Ri

_{*}), we conjecture that the melt rate follows Ri

_{*}as

## 8. Conclusions

We have evaluated direct numerical simulations against laboratory experiments of turbulent convection beneath a horizontal ice–water interface. The structure of the flow differs insignificantly between the laboratory experiment and the simulations, and the melt rates agree quantitatively. We have thus complemented the temperature profile of the laboratory experiment with the temperature and flow fields of the simulation. Based on this data, we have derived, from first principles, an analytical expression for the melt rate of ice *w*_{f} under purely thermally driven conditions.

The simulations show that molecular diffusion sets and limits the energy exchange at the ice interface for all far-field temperatures. For far-field temperatures below 8°C, a stably stratified diffusive layer shields the ice from the turbulent outer layer. The thickness of the stably stratified layer is *z*_{0} ∝ (*κ*^{2}/|*b*_{m}|)^{1/3}, with a buoyancy anomaly *b*_{m} = *b*(*T* = *T*_{m}), *T*_{m} = 3.98°C, and buoyancy *b* defined with respect to the density in the far field. The flow structure is similar to that seen in cloud-top mixing layers. With increasing far-field temperature *T*_{∞}, the stably stratified shield diminishes. For far-field temperatures far above 8°C, the stable stratification has effectively disappeared. Then the flow is similar to that found in free convection over a heated plate but upside down, with a diffusive layer thinner than the stably stratified diffusive layer.

The understanding of the flow structure is used to quantify the energy exchange at the ice–water interface. We provide in Eq. (21) an explicit parameterization of *w*_{f} that only depends on *T*_{∞}, and we give *w*_{f} as a function of Ri_{*} in Fig. 9b. The Richardson number Ri_{*} = *b* = *b*(*z* = 0) − *b*_{m} is defined with the buoyancy at the interface *b*(*z* = 0).

Based on the dependence of the melt rate on the far-field temperature *T*_{∞}, three regimes can be distinguished: a linear dependence in the diffusive regime Δ*T*_{∞} < Δ*T*_{m}, a dependence *w*_{f} ∝ (*T*_{∞} − *T*_{m})^{5/3} in the high temperature limit *T*_{∞} ≫ 8°C, and a dependence *w*_{f} ∝ (*T*_{∞} − *T*_{m})^{4/3} in between. We find that the influence of meltwater on the melt rate can be neglected for far-field temperatures *T*_{∞} < 20°C.

In contrast to previous work, we find a turbulent enhancement of the melt rate by one to two orders of magnitude for a given boundary layer depth. Hence, our findings imply that turbulence needs to be considered in the analysis of ice–water dynamics even in shear-free conditions to obtain accurate ice-bottom melt rates.

## Acknowledgments

We thank Alberto de Lozar and Cedrick Ansorge for interesting discussions on the topic. Laboratory assistants involved in data acquisition were Simon Kamprath, Felix Matt, Thilo Klenz, and Niels Fuchs. We also acknowledge the support by the institute’s workshop and Leif Riemenschneider. Computational resources were supplied by Jülich Supercomputing Centre and German Climate Computing Centre. Primary data and scripts used in the analysis and other supplementary information that may be useful in reproducing the author’s work are archived by the Max Planck Institute for Meteorology and can be obtained by contacting publications@mpimet.mpg.de.

## APPENDIX

### The Effective Buoyancy Flux

*B*

_{eff}is known a priori and is then used in combination with the boundary layer height

*z*

_{*}to construct a velocity scale from (

*B*

_{eff}

*z*

_{*})

^{1/3}(Deardorff 1970). In this work, however, a relevant buoyancy flux scale is not known a priori, nor is it readily evident from the internal-energy flux

*H*

_{water}via a buoyancy parameter that relates temperature and buoyancy [Eq. (5)]. Hence, we define an effective buoyancy flux byFor researchers interested in similar systems, we document that

*B*

_{eff}ranges between 0.1 and 0.4 in units of the diffusion-dominated buoyancy flux of free-convection systems

*θ*

_{m},

*B*

_{eff}decreases by a factor of 4 from the low temperature range to the high temperature limit, and

*B*

_{eff}is found to be constant in time, once the mixed layer is developed.

*H*

_{water}asThe linear relationship

*H*

_{water}/

*B*

_{eff}≈ 0.55

*θ*

_{m}

*T*

_{∞}/

*b*

_{m}is empirically determined. With Eqs. (5), (7), (A1), and (A2), it yieldsThe term

*H*

_{water}is thus obtainable for systems of known far-field temperature

*T*

_{∞}once estimates for the boundary layer height

*z*

_{*}and the convective velocity

*w*

_{*}are provided.

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