Aspect ratio affects iceberg melting
Eric W. Hester, Craig D. McConnochie, Claudia Cenedese, Louis-Alexandre Couston, Geoffrey Vasil
AAspect ratio affects iceberg melting
Eric W. Hester ∗ Craig D. McConnochie † Claudia Cenedese ‡ Louis-Alexandre Couston §¶ Geoffrey Vasil ∗ September 23, 2020
Abstract
Iceberg meltwater is a critical freshwater flux from the cryosphere to the oceans. Globalclimate simulations therefore require simple and accurate parameterisations of icebergmelting. Iceberg shape is an important but often neglected aspect of iceberg melting.Icebergs have an enormous range of shapes and sizes, and distinct processes dominatebasal and side melting. We show how different iceberg aspect ratios and relative ambientwater velocities affect melting using a combined experimental and numerical study. Theexperimental results show significant variations in melting between different iceberg faces,as well as within each iceberg face. These findings are reproduced and explained with novelmultiphysics numerical simulations. At high relative ambient velocities melting is largeston the side facing the flow, and mixing during vortex generation causes local increases inbasal melt rates of over 50%. Double-diffusive buoyancy effects become significant whenthe relative ambient velocity is low. Existing melting parameterisations do not reproduceour findings. We propose improvements to capture the influence of aspect ratio on icebergmelting.
Iceberg meltwater provides an important flux of freshwater from ice sheets to oceans [31, 25],making up 45% of Antarctic freshwater loss [28], and dominating freshwater production inGreenland fjords [13]. Melting also releases nutrients that boost biological productivity andcarbon sequestration [32]. Where and when meltwater and nutrients are released dependson how quickly icebergs melt. Understanding how icebergs influence the climate thereforerequires accurate predictions of iceberg melt rates. We present an experimental and numericalinvestigation of an often neglected aspect of melting — iceberg shape.Icebergs display enormous variation in shape and size [7, 33, 34, 12, 31, 2]. Horizontalextents range from several meters to the record iceberg B-15 at 300 km ×
40 km [3]. Depthsvary considerably but almost never exceed 600 m [10]. Rolling instability further constrainsrealistic shapes [35, 6]. Icebergs can tumble when the aspect ratio , the ratio of length L tosubmerged depth D , is smaller than (cid:112) .
92 + 58 . /D , where D is expressed in metres. Aspectratios may therefore range anywhere from 1 to 1000. The overall melting will depend stronglyon aspect ratio whenever bottom and side melt rates differ. ∗ University of Sydney School of Mathematics and Statistics, Sydney, Australia † Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, NZ ‡ Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA § British Antarctic Survey, Cambridge, UK ¶ Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France a r X i v : . [ phy s i c s . f l u - dyn ] S e p sing empirical relations for turbulent heat transfer over a flat plate [11], Weeks and Camp-bell developed a commonly used parameterisation for iceberg melt rates [35] (hereafter the WCmodel). The WC model predicts an iceberg melt rate v (in dimensional units of speed) of v = 0 . (cid:18) ρ w ρ i ν − / κ / c p Λ (cid:19) U . ∆ TL . . (1)Here, ρ w and ρ i are the respective densities of water and ice, ν and κ are the respectivediffusivities of momentum and temperature, c p is the heat capacity at constant pressure ofseawater, Λ is the latent heat of ice melting, U is the relative speed between the ambient waterand the iceberg, L is the iceberg length, and ∆ T is a characteristic temperature difference.Unfortunately the model makes several incorrect predictions, such as predicting that meltingstops for zero relative velocity. This shortcoming was addressed recently by including the effectof meltwater-plume entrainment for low relative ambient velocities [14]. More importantly, themodel fails to treat side and base melting separately, which are expected to be dominated bydifferent dynamical processes [6].Our goal is to understand the effect of aspect ratio on iceberg melting in a series of labora-tory experiments and numerical simulations. We compare our findings with predictions of theWC model and suggest improvements to account for the influence of aspect ratio on melting.In section 2 we describe the experimental method, and summarise the findings in section 3.Using a recent numerical method [19], summarised in section 4, we reproduce the laboratoryexperiments in a series of fluid-solid simulations, which allow us to identify and discuss keyphysical processes controlling melting in section 5. We summarise geophysical implicationsand discuss possible improvements to parameterisations in section 6. We conclude and discussfuture directions in section 7. Figure 1: ( a ) Experimental schematic. A dyed ice block of dimensions with length L , width W ,and immersed depth D is fixed in a flume with a relative ambient velocity U , temperature T w and salinity C w . ( b ) Post-experiment photographs (top) were filtered and reoriented (waterlinein red) to estimate the average melting of each face.The experiments immersed ice blocks of different lengths in a recirculating salt water flumewith different ambient water velocities (fig. 1 ( a )). The central section of the flume measured76 . . C w =30 g / kg and temperatures T w = 18 to 21 ◦ C. The 26 distinct experiments held the depth fixedat D = 3 cm. We considered 5 lengths L = 10 , , , ,
33 cm and three relative ambient flow2igure 2: Time series of two experiments with L = 10 cm (left) and L = 32 . D = 3 cm, in ambient water moving at U = 3 . − . Each frame isseparated by two minutes.velocities U = 0 , .
5, or 3 . − . The width varied from W = 10 to 22 . T i = − ◦ C. Theice block was weighed, positioned in the flume, and each experiment ran for 10 minutes. Theice block was then removed, reweighed to calculate mass loss, and photographed from each side.Post-experiment photographs determined the melting of each face (fig. 1 ( b )). A blue filterhighlighted the block, then thresholding returned a binary image, and an opening transformremoved the noise. The edges of the waterline (red line in fig. 1 ( b )) are defined as the lowestpoints with less than 15 pixels of melting, and the images were rotated to level the waterline.Pixels were converted to cm using rulers in each photo. The left profile was calculated as afunction of depth, and the left face defined as the portion with slope greater than 1. The depthaverage of the left face determined the average melt rate of that face. Average bottom andright melt rates were calculated similarly. High relative ambient velocity — Figure 2 shows a time series of two experiments at U = 3 . − . The melting on both front faces (left side of each frame) is much larger than onthe base and sides. This result agrees with previous studies showing that flow perpendicular toan ice face leads to greater heat transport and melt rates than flow parallel to an ice face [24].We therefore expect aspect ratio to influence melting at large relative ambient flow speeds.The second apparent feature is the non-uniformity of melting within each face of eachexperiment (fig. 2). The front melting increases with depth, decreasing the slope of the leadingface over time. The basal melting also has a pronounced non-uniform profile. Starting fromupstream, a darker region of increased dye concentration is pooled just behind the front ofthe base, and the melt rate is low. This concentration suggests a stagnant zone that doesnot mix with the incoming flow, which is typical when flow separation occurs. Immediately3igure 3: Time series of two L = 32 . U = 1 . − (left) and U =0 cm s − (right).behind this region we observe increased turbulence and basal melt rates in both experiments.At the rear the longer ice block (fig. 2, right) returns to a regime of lower, uniform melting.The dye patterns toward the rear of the longer block (right) become less mixed, suggesting lessturbulent flow. The melting pattern is similar for all blocks, and echoes understanding thatother liquid-solid phase change problems can evolve to self-similar shapes [17, 26, 22].The flow field helps explain the basal melting pattern. As fluid moves past a forward-facingstep, vorticity separates from the leading edge. This configuration produces a region of unsteadyrecirculation and subsequent reattachment of the flow. The few studies examining heat transferin flow past a forward-facing step find a maximum in convective heat transfer at the point ofreattachment [1, 23]. Different estimates for the reattachment length exist (summarised in[30]) but all find that it is roughly 3 to 5 times the step height for Reynolds numbers Re = 10 to 10 (our experiment is at Re ≡ U D/ν = 800, based on the step height). Actual icebergsin Greenland fjords may experience Reynolds numbers up to 2 × , assuming a draft of200 m, and local relative velocities up to 0 . − [15]. We believe some kind of turbulent-flowseparation-reattachment scenario to continue to much higher Reynolds numbers. We, therefore,expect to see similar behaviour in real icebergs.The free surface behind the large block is much darker than for the short block, suggesting alarger amount of meltwater is reaching the surface. Turbulent flows underneath the blocks helpexplain this observation. The flow underneath the shorter block is entirely turbulent, thoroughlymixing the meltwater. The flow underneath the longer block is instead more laminar, meaningmost meltwater does not mix with the ambient water and rises to the free surface. The increasedturbulence at the rear of the short block also explains the greater melt rate of the rear face. Low relative ambient velocity — The left column of fig. 3 shows a time series froman experiment performed at 1 . − . Many of the same trends are apparent from the U =3 . − experiments. The basal melt is at a maximum behind the leading (left) edge, followedby a return to laminar flow and more uniform melt rate further downstream. The stagnantregion size is comparable to that in the higher relative velocity experiments, which supports a Reindependent scaling of reattachment length [30]. The primary difference between the U = 1 . . − experiments is lower overall melting and reduced turbulence (dye streaks appearmostly laminar). Meltwater preferentially pools near the free surface, as there is less mixing4ith ambient water than at higher relative velocities U (fig. 2). No relative ambient velocity — The zero relative velocity experiments lack any localincreases in the melt rate. Most meltwater flows slowly along the base of the ice block beforerising to the surface. However some dyed fluid sinks from the base of the ice as a dense plume.This is best understood as a double-diffusive effect [16]. During the melting, the adjacent saltwater absorbs latent heat. The cooling occurs faster than salinity diffusion. With weak ambientflow, the saltwater can cool enough to sink and entrain dyed meltwater along with it. U = 3.5 cm s Final side profiles cm D e p t h z c m U = 1.5 cm s x cm0123 U = 0.0 cm s Figure 4: The top, middle, and bottom figures plot final side profiles of each experiment relativeto the bottom of the front face at the start of each experiment (origin), for respective ambientvelocities U = 3 . , . , − . Profiles are coloured according to initial aspect ratio. Thedashed lines show an averaged profile over all experiments at a given relative ambient velocity U . The approximate edges of each face are shown with circles. The grey bar illustrates astandard deviation in average total basal melt after 10 minutes calculated from table 1.Figure 4 plots the post-experiment profile of each ice block, and supports previous obser-vations. Melt rates differ between faces, with lowest melting on the base, and highest meltingon the sides. Larger ambient velocities U cause larger melt rates, particularly for the frontface. Significant non-uniformities in melt rate exist on each face. These non-uniformities arethe same for different experiments. Front and basal profiles at each relative ambient velocity U collapse on to a single average profile (dashed) for all initial aspect ratios.Figure 5 averages over the non-uniform melting of each face. The first three panels plotthe average melt rate of each face for each experiment at relative ambient velocity U = 3 . , . . − , as a function of initial aspect ratio. These calculations, derived from post-experiment photographs, quantify previous qualitative observations. Estimated melt rates foreach face averaged over all experiments at a given relative ambient velocity U are shown in solidlines in fig. 5. The uncertainties (coloured bars) are defined as twice the standard deviationof the melt rates. These statistical summaries are reproduced in table 1. The final plot infig. 5 compares the estimated volume loss for each experiment assuming average melt rates(table 1) with the measured volume loss from weighing before and after experiments. The5 L / D M e l t r a t e c m m i n U = 3.5 cm s L / D M e l t r a t e c m m i n U = 1.5 cm s L / D M e l t r a t e c m m i n U = 0.0 cm s Front melt rate v f Rear melt rate v r Side melt rate v s Basal melt rate v b V cm E s t i m a t e d V c m Volume loss estimate
Figure 5: The first three panels show estimated average melt rates of each face for each ex-periment, for each ambient velocity U . The solid lines plot the average melt rate for each faceaveraged over all experiments. The uncertainties (coloured regions) are twice the standard de-viation in average melt rates for each side. The final panel compares the volume loss estimatedfrom average melt rates (table 1) with the volume loss measured by weighing. Exact agreementis indicated with the dashed line, and the uncertainties in the estimated volume loss are therange of volume changes using high and low estimates for each melt rate.photo estimates are close to an exact correspondence (dashed), supporting the averages oftable 1.Velocity Front melt rate Side melt rate Rear melt rate Basal melt rate WC melt rate U cm s − v f cm min − v s cm min − v r cm min − v b cm min − v cm min − . . ± .
06 0 . ± .
05 0 . ± .
07 0 . ± .
04 0 . − . . . ± .
04 0 . ± .
04 0 . ± .
07 0 . ± .
018 0 . − . . . ± .
04 0 . ± .
04 0 . ± .
04 0 . ± .
016 0Table 1: Average melt rates of each face for each flow speed U from fig. 5. The uncertaintiesare twice the standard deviation in average melt rates in fig. 5. The WC model melt rates aregiven for upper and lower lengths L max = 32 . L min = 10 cm, with ρ i = 0 . − , ρ w = 1 .
021 g cm − , ν = 1 . × − cm s − , κ = 1 . × − cm s − , c p = 4 .
182 J g − ◦ C − ,Λ = 334 J g − , and ∆ T = 20 ◦ C, appropriate for ambient water at 20 ◦ C [4].The second through fifth columns of table 1 give experimental melt rates for the front, side,rear and basal melt rates respectively, at each relative velocity U . The melt rate of each facediffers at all relative velocities U . Aspect ratio can therefore affect overall melting by changingthe relative areas of these faces. The melt rate of each face increases slightly from U = 0 cm s − to U = 1 . − , and significantly from U = 1 . − to the more turbulent 3 . − experiments. This result agrees with [14], which observed roughly constant melt rates below athreshold relative fluid velocity of 2 . − . Though FitzMaurice et al.’s findings were basedon dominant side melting, and so are not directly applicable to basal melting, the existence ofa similar threshold velocity for basal melting is likely.We give estimated melt rates using the WC model in the rightmost column of table 1.The upper and lower values are for the longest ( L = 32 . L = 10 cm)block lengths. The WC model underestimates melt rates of all faces at all velocities, and Note that we do not use an average internal ice temperature of − ◦ C (the midpoint of the meltingtemperature 0 ◦ C and freezer temperature − ◦ C), as was done in [14]. Using an internal ice temperatureimplies that colder blocks will melt faster. This contradicts the fact that colder blocks reduce the heat fluxdifference at the interface, and therefore melt more slowly. We instead use the physically justified meltingtemperature of the interface. U = 3 . − . The updated model of [14]accounts for larger melt rates at lower velocities, however it is based on side plumes that are notapplicable to basal melting. The WC model is also unable to capture the elevated melt ratesof the vertical faces. An accurate parameterisation must therefore account for both magnitudeand orientation of relative ambient flow. Existing melting parameterisations do not agree with our experimental findings. We use DirectNumerical Simulation (DNS) of melting ice in warm salty water to investigate the full flowdynamics and further our understanding of our laboratory observations. We simulate thischallenging problem with a recent phase-field approach developed for coupled fluid flow, melting,and dissolution [19]. This method builds on previous methods for fluid-solid interactions [20]and simulations of melting in fresh water [9, 27].
Ice melting in salt water is often modelled as a moving boundary problem [37]. In the fluid,the temperature T and salinity C satisfy advection-diffusion equations, and the fluid velocity u and pressure p satisfy incompressible Navier-Stokes equations with vertical buoyancy forcing − gρ ( T, C )ˆ z . The solid temperature follows a diffusive equation. Boundary conditions at themelting interface complete the system. The temperature is continuous and equal to the meltingtemperature, and Stefan, Robin, and Dirichlet boundary conditions conserve energy, salt, andmass respectively.Phase-field models are a smoothed approximation of the moving boundary formulation thatis physically motivated and simple to simulate [5]. Distinct phases are represented with asmooth phase field φ that is forced to φ ≈ φ ≈ ε separates the fluid and solid. The phase-field equations augment the bulk equationswith smooth source terms that reproduce the boundary conditions in the limit ε → ε
56 Λ c p κ ∂ t φ − γ ∇ φ + 1 ε φ (1 − φ )( γ (1 − φ ) + ε ( T + λC )) = 0 , (2a) ∂ t T + ∇ · ((1 − φ ) uT − κ ∇ T ) = Λ c p ∂ t φ, (2b) ∂ t ((1 − φ + δ ) C ) + ∇ · ((1 − φ + δ )( uC − µ ∇ C )) = 0 , (2c) ∂ t u + u · ∇ u − ν ∇ u + ∇ p + gρ ( T, C ) ρ ˆ z = − νη φ u, (2d) ∇ · u = 0 . (2e)Here ν, κ, and µ are momentum, thermal, and salt diffusivity, Λ is the latent heat, c p is theheat capacity of water, and λ is a slope coefficient of the liquidus, which is assumed linear forsimplicity [19]. The damping time η (cid:28) γ (cid:28) δ (cid:28) ε [19].Note that this model omits second order thermodynamic effects. We ignore density changesduring melting, consider constant viscous, thermal, and solutal diffusivities, and describe buoy-ancy with the Boussinesq approximation (using the EOS-80 equation of state of seawater). We7lso use two-dimensional simulations to reduce computational costs. While the experimentsdid not vary much in the spanwise direction orthogonal to the flow, the reduced dimension isknown to generate larger vortices than in three dimensions. We perform two series of simulations. The first series considers large relative ambient velocity U = 3 . − , and compares a simulation with temperature, salinity, and buoyancy effectsturned on ( T + C ) and a simulation with salinity and buoyancy forcing neglected ( T only). Thesecond series investigates double-diffusive effects with no relative ambient velocity U = 0 cm s − by comparing simulations with equal temperature and salt diffusivities (single diffusion: SD)and different temperature and salt diffusivities (double diffusion: DD). All simulations specifythe initial ice temperature as T i = 0 ◦ C, the initial water temperature as T w = 20 ◦ C, theinitial salinity as C w = 30 g kg − , the liquidus slope as λ = 0 . ◦ C kg g − , the latent heatas Λ = 3 . × J g − , and the heat capacity as c p = 4 . − ◦ C − . The relative ambientvelocity U , domain dimensions (cid:96) × d , block dimensions L × D , and diffusivities ν, κ, µ are givenin table 2. We use a realistic Prandtl number Pr = ν/κ of 7. Due to computational constraints,we are limited to larger salt diffusivities than reality, so the Schmidt number Sc = ν/µ is atmost 50 rather than 500.All simulations are performed using the spectral code Dedalus [8]. The equations are dis-cretised with Fourier series in the horizontal direction, and trigonometric series in the verticaldirection. This corresponds to periodic horizontal boundary conditions; homogeneous Neumannvertical boundary conditions for the horizontal velocity u , temperature T , salinity C , pressure p , and phase field φ ; and homogeneous Dirichlet vertical boundaries for the vertical velocity w .The simulations at U = 3 . − use a volume penalised ‘sponge layer’ at the beginning of thedomain to force the fluid temperature, salinity, and velocity to ambient values [20]. The parityconstraint of the buoyancy term in the vertical momentum equation is enforced by taperingbuoyancy near the vertical boundaries over a length scale of 0 .
05 cm. The system is integratedusing a second order implicit-explicit Runge-Kutta timestepper. The horizontal mode number n x , vertical mode number n z , time step size ∆ t and numerical phase field parameters are givenin table 3. All code is available online [18].Table 2: Model parameters for flow and no-flow simulations.Simulation U (cid:96) d L D ν κ µ cm s − cm cm cm cm cm s − cm s − cm s − Flow T + C . × − . × − . × − Flow T only 3.5 50 15 30 3 1 . × − . × − N/ANo-flow DD 0 20 10 10 3 1 . × − . × − . × − No-flow SD 0 20 10 10 3 1 . × − . × − . × − Table 3: Numerical parameters for flow and no-flow simulation series.Simulation n x n z ∆ t ε γ η δ series s cm cm ◦ C sFlow 6144 1536 1 . × − . × − × − No-flow 2048 1024 5 × − . × − × − Computational results
The first series simulates the experiments at high relative ambient velocity U = 3 . − .The T + C simulation uses the equation of state of seawater (EOS-80) to capture salinity andbuoyancy effects from experiments, while the T only simulation ignores these effects.Figure 6 shows a time series of temperature in the T + C and T only simulations. Both sim-ulations display similar patterns and reproduce several qualitative features of the experiment.The largest melting occurs on the front (left) face, where warm ambient water collides with theice. The front melting increases with depth, causing a decrease in slope over time. The basemelts most rapidly at the centre, but is overall slower than the front. These features are presentin both simulations and therefore cannot be controlled by the buoyancy of the meltwater. Athigh ambient velocities heat transported by the flow determines melting.Figure 7 quantifies the front, basal, and rear melting for each simulation. The front meltingis steady in both simulations (fig. 7 top left), with little deviation between instantaneous andcumulative melt rates (fig. 7 bottom left), and an increase in front melting with depth (fig. 7bottom left). The T + C simulation reproduces the sloped front of experiments, while the T only simulation has a vertical gradient at the top of the front. This difference in slope occursbecause cool, buoyant meltwater pools at the top of the T + C simulation, thickening the thermalboundary layer and slowing melting. Despite this difference, both simulations reproduce theexperimental average front melt rate of 0 . ± .
06 cm min − , with 0 . ± .
14 cm min − for the T + C simulation, and 0 . ± .
09 cm min − for T only (fig. 7 centre left).The basal melting shows significant variation in space and time (fig. 7 top middle). Bothsimulations show the same distinct regions of basal melting. Just behind the leading edge,melt rates are low and steady. This stagnant region is evident in the time series plots of fig. 6.Limited mixing with the warm ambient water causes reduced melting near the leading edge.Behind this region, pooled meltwater becomes unstable due to the strong velocity shear, leadingto vortex generation and shedding around the centre of the block. The unsteady flow circulateswarm ambient water to the base, and is associated with the transition to larger melt rates pastthe centre of the base. Beyond this point the melting is on average lower, with intermittentperiods of high melting occurring as vortices are shed downstream. The transition to highermelting slowly moves backward over time. This is partly because the front itself is receding,due to melting. More importantly the slope of the front face reduces over time, delaying theinstability of the shear layer generated at the separation point. However the region of maximumcumulative melting does appear to saturate around x = 28 cm at late times.The localised spatial and temporal features of the basal melting lead to nontrivial sta-tistical properties, summarised in histograms of instantaneous melt rates in the second rowof fig. 7. There is large variance and skewness in the basal melt rates, as expected fromthe different melting regions. The average basal melt rates of the simulations ( T + C :0 . ± .
08 cm min − , T only : 0 . ± .
08 cm min − ) agree with the upper range of experimentalresults (0 . ± .
04 cm min − ), supporting the validity of the simulations.The time averaged cumulative melt rates are given in row three, and clearly reproduce thelocalised increase in basal melting from the experiments. The cumulative basal melt varieswith distance by a factor of two, highlighting the importance of localised flow features in melt-ing predictions. The location of maximum melting is further downstream than observed inexperiments, however this may be due to small domain size, or two rather than three dimen-sional turbulence. Nevertheless it is clear that the buoyancy plays little role for large ambientvelocities. It is advection of heat that drives melting.Figure 8 illustrates the mechanism by which the flow induces melting hot spots. The two leftfigures show snapshots of the temperature, vorticity, and velocity at t = 75 s. The temperature9igure 6: Time series, at one minute intervals, of temperature for melting simulations in warmsalt water at relative ambient velocity U = 3 . − . The T + C simulation (left) includestemperature, salinity, and buoyancy, while the T only simulation (right) omits salinity andbuoyancy effects.Figure 7: The left, middle, and right columns summarise front, base, and rear melting respec-tively for U = 3 . − simulations ( T + C and T only). The front is defined where theinterface slope is less than −
1, the base where the slope is between − T + C simulation at 75 s. The vorticity plot also shows the fluid velocity (black arrows).The melt rate (solid red line) is enhanced where vortices generate upwelling. The right panelplots the basal melt rate over space and time. Vortex locations (grey) are shown where thedepth integrated vorticity is less than −
2. The time t = 75 s is shown in black.field shows a cool stagnant region that persists past the separation point at the front edge.Mixing between this region and the ambient fluid is slow and intermittent, and the melt rate(red curve) remains low. The vorticity plot shows prominent vortices being generated behindthe stagnant region due to Kelvin-Helmholtz instability of the shear layer. The fluid velocity (inarrows) reveals upwelling downstream from the vortices, which coincides with increased basalmelting (red curve). This explains both the mechanism by which vortices enhance melting –upwelling of warm ambient water – and accounts for the offset between vortex position andenhanced melting.The right panel of fig. 8 shows the instantaneous basal melt rate over space and time. On topof this plot, we highlight the location of vortices in grey by thresholding the depth integratedvorticity. This plot clarifies the mechanism by which vortices enhance melting. Basal melt ratesare low (roughly 0 . − ) in the stagnant region where there are no vortices to generatemixing. Behind this region vortices are being created. These vortices transport warm ambientwater to the base of the ice by upwelling to their right. This is clear from the large melt rates(almost 0 . − ) to the right of the vortex paths (grey stripes) in fig. 8. The vortices thenreach a maximum size and are rapidly advected away by the flow. In this stage they are nolonger close enough to the base to enhance melting and the melt rate returns to a lower averagevalue of around 0 . − . We therefore do not expect a recurrence of localised meltingwithout a new source of instability to generate vortices and enhanced mixing.The rear melting also shows noticeable variation in space and time (fig. 7 top right), with animportant difference from experiments. While the rear face in experiments sloped to the right(fig. 4), the rear face of the T + C simulation is on average vertical, and the T only rear face slopestoward the left (fig. 7 bottom right). The incorrect sloping of the rear face in the simulationsis likely due to large coherent vortices that remain behind the block, circulating warm ambientwater toward the rear face from above (fig. 6). It is possible these coherent vortices are two-11imensional features which would break apart in three dimensions. However, average rear meltrates are similar to the experimental value of 0 . ± .
07 cm min − , with 0 . ± .
13 cm min − for the T + C simulation, and 0 . ± .
11 cm min − for the T only simulation. The next simulation series investigates melting in an initially quiescent fluid ( U = 0 cm s − ).Experiments showed some meltwater sinking beneath the block, despite the high salinity of theambient water. We show how double-diffusivity causes this sinking by comparing a double-diffusive (DD) simulation with different thermal and salinity diffusivity (Lewis number Le = κ/µ = 50 /
7) to a single-diffusive (SD) simulation with equal diffusivities (Le = 1). TrueLewis numbers Le are much larger for salt water, however these simulations are enough todemonstrate the existence of double-diffusive effects. Both simulations use identical buoyancyfunctions (EOS-80 of seawater) with equivalent dependence on temperature and salt.Figure 9 plots time series of temperature and salinity of the two simulations of melting icein initially stationary salt water. Both simulations show a clear tendency of meltwater (darkred/purple) to rise and pool near the free surface, reducing melting in that region at late times(fig. 10 top left). But there are important differences between the simulations. The layer ofmeltwater is more diffuse for equal diffusivities (SD). And it is only with different diffusivities(DD) that sinking plumes emerge beneath the ice block. This confirms the double-diffusiveorigin of sinking plumes in experiments. Initially the sinking plumes are concentrated at thesides of the ice block, where the geometry aids the instability. However the sinking tends overtime to a persistent downward plume beneath the centre of the ice block (as in experiments).In contrast, no sinking plumes occur beneath the ice block for the equal diffusivity simulation.Some sinking is observed beneath the pooled meltwater, but this is because the rising waterbeneath the block sets up a recirculating flow that sinks and entrains meltwater near (periodic)horizontal boundaries. All meltwater around the ice block rises without double-diffusion. Thisleads to large differences in flow patterns, affecting melt rates.Figure 10 quantifies the melting of the two simulations. The first row plots a colour timeseries of the instantaneous melt rates on the side and base of each block. Fine-scale localisedmelting behaviour occurs on all faces in the double-diffusive simulation, whereas the equaldiffusivity simulation shows little spatial or time variation. This difference follows from theintermittent plumes generated via double diffusion. The second row of fig. 10 shows that thesedistributions result from processes with nontrivial spatial and temporal structure, and are notsimple Gaussian processes. The mean and standard deviation of melt rates are larger for bothfaces for the double-diffusive simulation (DD). The double-diffusive simulation has a side meltrate (0 . ± .
10 cm min − (DD)) that is closer to experimental melt rates (0 . ± .
04 cm min − )than the equal diffusive simulation (0 . ± .
08 cm min − (SD)). The slight underestimate of sidemelting is an understandable consequence of the restricted domain size, which causes poolingmeltwater to reduce side melt rates at late times. The basal melting of the double-diffusivesimulation (0 . ± .
10 cm min − (DD)) and equal diffusivity simulation (0 . ± .
02 cm min − (SD)) are both close to experimental basal melt rates (0 . ± .
016 cm min − ). In the final row(fig. 10) we give the cumulative melt rates, which show faster side melting for the DD simulationat early times, which become slower at late times due to pooled meltwater. Both simulationspredict larger side melt rates than basal melt rates, as in experiments. The difference in meltrates and flow patterns demonstrate the potential importance of double-diffusive effects foricebergs at low relative flow velocities. 12igure 9: Time series, at one minute intervals, of temperature (left) and salinity (right) forsimulations in quiescent salt water. The left simulation in each column uses a lower saltdiffusivity (DD), while the right simulation in each column has equal salt and thermal diffusivity(SD).Figure 10: The left and right columns summarise side and basal melting respectively for the norelative ambient velocity simulations. The first row shows instantaneous melt rates over spaceand time. The second row givens a histogram, average, and standard deviation of instantaneousmelt rates. The final row plots cumulative melt rate during the simulation. The right face isanalogous to the left face and omitted. 13 Geophysical application
The laboratory and numerical results highlight that melt rates vary on each ice block face, andthat different faces have different average melt rates. The average melt rates of each face arealso significantly higher than predicted by commonly used parameterisations. These differencesin melt rates matter for melting of real icebergs. L / D = W / D M e l t r a t e v a v c m m i n v b = 0.13 cm min v b = 0.080 cm min v b = 0.077 cm min Geometrically averaged melt rate v a cm min U = 3.5 cm s U = 1.5 cm s U = 0.0 cm s Figure 11: Illustration of geometry-weighted melt rates v av as a function of aspect ratio, foreach relative ambient velocity U , using average experimental melt rates for each face (table 1).The basal melt rate v b for each velocity is plotted in dashed lines, and annotated on the plot.We expect melt rates to differ between faces for real icebergs, as different faces will beexposed to different fluid velocities, and experience different buoyancy effects. Even relativelylow velocities towards an ice face cause higher melt rates than velocities parallel to the ice face[24]. Different melt rates for different faces should occur even without relative ambient flows,as buoyancy driven plumes induce larger velocities and melt rates on the sides [14].Aspect ratios of geophysical icebergs also vary significantly, affecting the relative importanceof melt rates on each face. The B-15 iceberg had an estimated length of 300 km and width of40 km [3]. Using a depth of 600 m as an upper limit [10] suggests a minimum aspect ratio ofapproximately 180 (using the geometric mean of the two horizontal length scales). At the otherextreme, stability suggests a minimum aspect ratio of 1.4 for a 50 m deep iceberg. This meansthe basal area of B-15 comprises 97% of the total submerged area while the basal area of asmaller marginally stable iceberg is only 27%. Different melt rates on each face will thereforematter for aspect ratios close to unity, whereas large aspect ratio icebergs will be dominatedby basal melting.We illustrate the influence of aspect ratio on melting using a simple geometric model ofa melting block of ice that assumes each face melts uniformly. The length L , width W , andsubmerged depth D are related to melt rates of the front v f , rear v r , side v s , and base v b via˙ L = − v f − v r , ˙ W = − v s , and ˙ D = − v b (denoting time derivatives with dots). The immersedvolume V and area A are V = LW D , and A = 2 LD + 2 W D + LW . The geometrically averaged melt rate v av normalises the volume loss rate by the surface area, v av ≡ A ddt | V | = ( v f + v r ) W DA + 2 v s LDA + v b LWA , (3)which weights the melt rate of each face by its relative proportion of the total area.Figure 11 plots the geometrically averaged melt rate v av as a function of aspect ratio L/D ,using average melt rates from experiments (table 1) and simplifying the horizontal dimensions14s equal ( W = L ). Figure 11 shows significant variation in the geometrically averaged meltrate v av from aspect ratio 1 to 50. Ice blocks with unit aspect ratio L = D melt more than 50%faster than large aspect ratio ice blocks, with elevated overall melting (relative to the dashedbasal melt rates) apparent even for aspect ratio 10.The toy model also offers a simple criterion for side-dominated versus base-dominated melt-ing. An iceberg will experience side-dominated melting if the aspect ratio decreases over time,and base-dominated melting if the aspect ratio increases over time. If we simplify to two di-mensions, length L and depth D , with respective melt rates v s and v b , then the time derivativeof the aspect ratio is ddt (cid:18) LD (cid:19) = ˙ LD − L ˙ DD = LD (cid:16) v s L − v b D (cid:17) (4)The time derivative of aspect ratio changes sign when the ratio of side to base melting v s /v b is equal to the aspect ratio L/D . Therefore even icebergs with larger side melting can becomedominated by basal melting if the aspect ratio is sufficiently large. Returning to three dimen-sions, if the melt rate differs between different side faces (as for the experiments with relativeambient velocity), different regimes can be defined for each orientation of the iceberg.We also expect non-uniform melt rates on each face to affect geophysical icebergs. Themost common iceberg length of tabular icebergs in the Southern Ocean is 400 m, and the mostcommon freeboard f is 35 m [29]. Using the empirical relationship D = 49 . f . [29] gives amost common depth D of approximately 100 m. Our experiments find a maximum in meltingat a distance of two and a half times the depth, similar to studies of heat transfer past aforward facing step [1]. This predicts a maximum in basal melting at approximately 250 m, asignificant proportion of the basal length of the modal tabular iceberg. This acts to furtherenhance overall melt rates for smaller aspect ratio icebergs. Of course, confirming that thisproportionality holds at larger scales, with ocean stratification, would require further work.Nevertheless studies of flow past a forward facing step find this length scale is a small multipleof the step height over a large range of Reynolds numbers [30]. Typical existing parameterisations of iceberg melting ignore the aspect ratio of icebergs [36, 21].We conducted a series of laboratory experiments and numerical simulations to examine thedependence of the melt rate on iceberg size and shape for three different ambient velocities.We find that geometry has a strong effect on the melt rate of icebergs.Melt rates are highest on the forward facing side (with respect to the ambient flow), fol-lowed by the remaining lateral sides, with slowest melting occurring at the base of the iceberg.Changing the relative area of each face will thus change the overall melt rate, with significantvariation between small and large aspect ratio icebergs.Furthermore, the melt rate of each face is itself spatially non uniform, with localised in-creases in basal melt rates of up to 50% observed. These localised regions correspond to thereattachment zones of flow past a forward-facing block, and occur at a distance approximatelytwo to three times the depth of the block [1]. Our numerical investigation showed that thesenon-uniformities in basal melt are caused by the generation of vortices, which lead to upwellingof warm water during their formation. This non-uniformity exists at large relative speeds, andis not influenced by buoyancy. We therefore expect these non-uniformities to increase meltingfor small-aspect ratio icebergs.To improve melting estimates, we emphasise that models of melt rates must depend onboth the speed and orientation of the background flow, and that differing melt rates must beweighted according to the shape and aspect ratio of an iceberg.15
Acknowledgments
Eric Hester is grateful for support from NSF OCE-1829864 during his 2017 Geophysical FluidDynamics Summer Fellowship at the Woods Hole Oceanographic Institution, as well as supportfrom the University of Sydney through the William and Catherine McIllarth Research TravelScholarship. Louis-Alexandre Couston acknowledges funding from the European Union’s Hori-zon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agree-ment 793450. We acknowledge PRACE for awarding us access to Marconi at CINECA, Italy.
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