Melt ponds on arctic sea-ice
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globals[ ;mean-ice-height is defined via the GUI ;smooth-cycles is defined via the GUI density-ratio ;melt-rate-ice ;melt-rate-pond water-height-transition ;seepage-rate gravity kinematic-viscosity horizontal-permeability albedo-sea-water albedo-ice ;; physical parameters for the model time-step space-step ; not used for the moment ;utility variables max1 max2 initial-mean-ice time-to-half-of-initial-mean-ice ] breed[drops drop] ; internal variables on the patches: ; "ice" is ice height in [cm] ; "water" is the melt water in [cm] ; "melt" represents the water recently melted in [cm] ; "albedo" the fraction of radiation reflected by the surface patches-own[ice water albedo] drops-own[water-content] ;; we affect the global physical parameters that are relevant for the model to startup set time-step 0.5 ; expressed in days set space-step 100. ; the lateral size of a patch expressed in cm ;set melt-rate-ice 1.2 ;in cm of ice per day ;set melt-rate-pond 2.0 ;in cm of ice per day, nominal value, check the literature set water-height-transition 10 ;in cm set density-ratio 0.8 ;ratio between ice and water mass densities , for less porous ice can be 0.9 ;ifelse seepage? [ ;set seepage-rate 0.8 ; in cm / day ;][ ;set seepage-rate 0.0 ; in cm / day ;] set gravity 9.81 * 100 * (86400 * 86400 ) ;in cm / day^2 set kinematic-viscosity 1.e-6 * (100 * 100) * 86400 ;in cm^2 / day set horizontal-permeability 3.e-9 * (100 * 100); in cm^2 set albedo-sea-water 0.1 set albedo-ice 0.9 set max1 0 set max2 0 set time-to-half-of-initial-mean-ice 0 end ; generate a random smooth topography to setup-topography if clear-previous-plots? [clear-all] startup ask patches [ set ice random-float (2 * mean-ice-height) set water 0 set albedo compute-albedo set pcolor scale-color cyan ice 0 (2 * mean-ice-height) ] ifelse(smooth-with-radius?) [ smooth-ice-with-radius][ smooth-ice-with-cycles] set initial-mean-ice compute-mean-ice reset-ticks tick ;this is to enable the plotting of the histogram of the initial configuration reset-ticks end ; this is the function to smooth the random ice field to smooth-ice-with-cycles repeat smooth-cycles [ ask patches [ let sum-ice-neighbors sum [ice] of neighbors4 let mean-ice (sum-ice-neighbors + ice) / 5 set ice mean-ice + random-float 0.1 color-field ] ] end ; this is the function to smooth the random ice field with a different procedure to smooth-ice-with-radius ask patches [ ;let sum-ice sum [ice] of patches in-radius smooth-radius ;let count-ice count patches in-radius smooth-radius ;let mean-ice (sum-ice + ice) / ( count-ice + 1 ) let mean-ice mean [ice] of patches in-radius smooth-radius set ice mean-ice ] ask patches[ color-field ] end ; coloring functions to color-field ifelse water > 0 [ set pcolor scale-color blue (2. * mean-ice-height * density-ratio - water) 0 (2. * mean-ice-height * density-ratio) ][ if ice > 0 [set pcolor scale-color grey ice -10 (2 * mean-ice-height)] ;same result as the following 2 lines ;let ice-color 80 + ((89.9 - 80) / (2 * mean-ice-height)) * ice ;set pcolor ice-color if ice = 0 [set pcolor turquoise] ;blue - 3] ] end ; melt ice to melt-ice let actual-melted-volume 0 if ice > 0 [ ;; VERTICAL MELTING ;; 1) the following line implements conductive melting of ice if water = 0 [ set actual-melted-volume melt-rate-ice * time-step ;shall be corrected to include Stefan effect, see line below ;set actual-melted-volume ( melt-rate-ice * time-step / ice * mean-ice-height ) ] ;; 2) the following line implements the increased melt-rate for pondend ice : water enahnces melting (as in Luethje et al. paper) if water > 0 [ ifelse water < water-height-transition[ set actual-melted-volume (melt-rate-ice + (melt-rate-pond - melt-rate-ice) * (water / water-height-transition) )* time-step ][ set actual-melted-volume melt-rate-pond * time-step ] ] ;; lateral melting if pond-lateral-melting?[set actual-melted-volume (actual-melted-volume + (melt-rate-pond * lateral-melting * time-step))] ;; Making melting happening if actual-melted-volume > ice [ set actual-melted-volume ice ;this is to avoid to melt more ice than what we have set water 0 ] set ice (ice - actual-melted-volume ) ; melt has occurred sprout-drops 1 [ set water-content (actual-melted-volume * density-ratio) ; the melted water is put into a pocket (a moving agent) ifelse pen-down? [pen-down][pen-up] hide-turtle ] ] end ;; seepage of meltwater to seepage let seepage-amount (seepage-rate * time-step) ; the seepage amount per patch ask drops-here [ ifelse water-content > seepage-amount[ set water-content water-content - seepage-amount set seepage-amount 0 ][ set seepage-amount (seepage-amount - water-content) set water-content 0 die ] ] if seepage-amount > 0 and water > 0[ ifelse water > seepage-amount[ set water (water - seepage-amount) ][ set water 0 ] ] end ;; move drops till conversion into water ;; flow assumes that the water produced by melt displaces instantaneously to position of minimum potential energy to flow loop[ let p min-one-of neighbors [ice + water] ifelse (ice + water) > [ice + water] of p [ move-to p ][ set water (water + water-content) ; the pocket of melted water has reached a minimum height and it remains there if ice = 0 [set water 0] if not melt-ponds? [set water 0] ; remove all water if we are not interested in ponds die ] ] end ; move drops till conversion into water ;; flow2 assumes that the spped is finite. It depends on the local gradient of "ice + water" and by the size of the patch. Needs improvement. to flow2 let p min-one-of neighbors [ice + water] ;let displacement (gravity / kinematic-viscosity) * (((ice + water) - [ice + water] of p) / space-step) * time-step ;let gradient (((ice + water) - [ice + water] of p) / space-step) let h atan space-step ((ice + water) - [ice + water] of p) let angle (90 - h) mod 360 let horizontal-velocity 2 ;1000 * space-step * sqrt (abs (gravity * (sin angle)) / (2 * space-step) ) let displacement (horizontal-velocity * time-step) ifelse displacement > distance p [ move-to p ][ set water (water + water-content) ; the pocket of melted water has reached a minimum height and it remains there if ice = 0 [set water 0] if not melt-ponds? [set water 0] ; remove all water if we are not interested in ponds die ] end to refreeze ask drops[ set water (water + water-content) ; the pocket of melted water has reached a minimum height and it remains there if ice = 0 [set water 0] if not melt-ponds? [set water 0] ; remove all water if we are not interested in ponds die ] ask patches[ set ice ice + (water / density-ratio) set water 0 color-field ] tick end ; a function for patches to account for the contribution to melting due to the presence of nearby ponds to-report lateral-melting let ice-here ice let melt-volume 0 ask neighbors4 with [ice < ice-here][ ;ask neighbors with [ice < ice-here][ ifelse (ice + water > ice-here)[ set melt-volume (melt-volume + (ice-here - ice)) ][ set melt-volume (melt-volume + water) ] ] set melt-volume (melt-volume / space-step) report melt-volume end ;; MAIN ;; this is the procedure for the loop over time to melt-and-flow if pen-down? [cd] ; to clear previous drawings ask patches[ ;; melt the ice melt-ice ;; seepage if seepage? [seepage] ] ask drops[ ;; move water if water-flowing-mode = 1 [flow] if water-flowing-mode = 2 [flow2] ] ask patches[ ;; recoloring the map color-field ;; estimate albedo set albedo compute-albedo ] if (count patches with [ice = 0] = count patches )[ stop ] tick end ; reports used in plots to-report compute-mean-ice report mean [ice] of patches end to-report compute-mean-water report mean [water] of patches end to-report compute-max-mean-water let water-depth mean [water] of patches if water-depth > max2 [ set max2 water-depth] report max2 end to-report compute-std-ice let avg compute-mean-ice let var (mean [ice * ice] of patches) report sqrt (var - ( avg * avg )) end ; we need to improve this, it's just for test to-report compute-albedo let alpha albedo-sea-water ; we start from total absorption (like if there is sea everywhere) if ice > 0 [ ifelse water > 0 [ let kappa (-1 / water-height-transition) * ln ( albedo-sea-water / albedo-ice ) set alpha albedo-ice * exp (- water / water-height-transition) ; albedo coeff of pond ][ set alpha albedo-ice ; albedo coeff of ice ] ] report alpha end to-report max-pond-coverage let value 100 * (count patches with [water > 0]) / count patches if value > max1 [set max1 value] report max1 end to-report time-to-50%-ice-melt if compute-mean-ice < 0.5 * initial-mean-ice and time-to-half-of-initial-mean-ice = 0 [ set time-to-half-of-initial-mean-ice (ticks * time-step) ] report time-to-half-of-initial-mean-ice end
There is only one version of this model, created over 6 years ago by Enrico Calzavarini.
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File | Type | Description | Last updated | |
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Melt ponds on arctic sea-ice.png | preview | Preview for 'Melt ponds on arctic sea-ice' | over 6 years ago, by Enrico Calzavarini | Download |
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