Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model

Modelling ice shelf basal melt with Glimmer-CISM coupled to a meltwater plume model Carl Gladish NYU CIMS February 17, 2010 Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 1 / 24

Acknowledgements Thanks to: David Holland (NYU) Paul Holland (BAS) Bill Lipscomb (LANL) Steve Price (LANL) Project supported by: Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 2 / 24

Contents 1 Motivation / Goals 2 Model description 3 Model Results 4 Model Equations (if anyone asks) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 3 / 24

Channelized basal melt on Petermann Glacier in NW Greenland. (Rignot & Steffen GRL 2008) Figure: Ice upper and lower surfaces Figure: Basal melt rate Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 4 / 24

Science Goals How do ice shelf melt water channels form? Are their wavelengths determined by bedrock or ocean interaction? What are they sensitive to? Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 5 / 24

Contents 1 Motivation / Goals 2 Model description 3 Model Results 4 Model Equations (if anyone asks) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 6 / 24

Coupled system: Ice-shelf, Melt Water Plume, Ocean Cavity Figure: From P.R. Holland and Feltham (JPO 2006) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 7 / 24

Ice model σ = ρg Glen s law rheology 3D advection of temperature, vertical diffusion, strain heating incompressible imposed accumulation Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 8 / 24

Plume model u, T, S equations z-integrated from A to B incompressible D sources: ė, ṁ DU sources: ρ, A, wall drag DT sources: ṁt B + ėt A, turbulent transfer DS sources: ės A Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 9 / 24

Coupling Variables Mass exchange: ṁ Geometric: A = H ρ i ρ o D Heat conduction (future) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 10 / 24

Entrainment where and ė = c2 l Sc T (U 2 + V 2 ) Ri = ( 1 + Ri ) + ė source Sc T g D U 2 + V 2 ė source enforces a minimum thickness of the plume Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 11 / 24

Typical t ice = 0.5y. Typical t plume = 60.0s. Coupling pseudo-code (shelf driver.f90) initialize models run plume to steady-state w.r.t. initial ice for each ice timestep: run ice timestep subcycle plume to steady-state or end of t ice Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 12 / 24

Contents 1 Motivation / Goals 2 Model description 3 Model Results 4 Model Equations (if anyone asks) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 13 / 24

Plume only - running to steady-state 10km x 10km, 25 days 100 by 100 grid, t = 60.0s 20 m minimum thickness 100m channel amplitude Uniform ambient ocean 1 C and 34.5psu 500 600 700 800 600 650 700 750 800 with and without rotation 900 1000 10000 8000 6000 4000 2000 Y (meters) 0 0 2000 6000 4000 X (meters) 8000 850 900 10000 950 1000 Figure: Ice shelf basal depth Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 14 / 24

Movie of plume runs to steady-state Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 15 / 24

Coupled Ice-Plume run 5km x 10km, 50 years 25x50 grid, t ice = 0.5 no-slip on North,East,West boundaries thk (meter) 0 100 200 300 400 500 (time, x1) = (0, 2400) thk (meter) 0 50 100 150 200 250 300 350 400 (time, y1) = (0, 4000) 0 2000 4000 6000 8000 10000 y1 (meter) ice thickness from coupled confined shelf test 0 1000 2000 3000 4000 5000 x1 (meter) ice thickness from coupled confined shelf test Figure: Initial thickness along flow Figure: initial thickness across flow Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 16 / 24

Basal melt rate bmelt (meters/sec)/10**-6 0 0.1 0.2 0.3 0.4 0.5 0.6 (time, x) = (1.59366e+09, 1200) (time, x) = (1.59366e+09, 1800) (time, x) = (1.59366e+09, 2600) 0 2000 4000 6000 8000 10000 bmelt (meters/sec)/10**-6 0 0.1 0.2 0.3 0.4 0.5 0.6 (time, y) = (1.59366e+09, 6400) (time, y) = (1.59366e+09, 4800) (time, y) = (1.59366e+09, 3200) (time, y) = (1.59366e+09, 2000) (time, y) = (1.59366e+09, 1200) 0 1000 2000 3000 4000 5000 y (meter) x (meter) basal melt rate basal melt rate Figure: Along-flow ṁ at 50 years Figure: Across-flow ṁ at 50 years Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 17 / 24

Movie of upper surface Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 18 / 24

Contents 1 Motivation / Goals 2 Model description 3 Model Results 4 Model Equations (if anyone asks) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 19 / 24

blue = neglected terms red = coupling terms Momentum equation (force balance) Constitutive law x σ xx + y σ xy + z σ xz = 0 x σ yx + y σ yy + z σ yz = 0 x σ zx + y σ zy + z σ zz = ρ ice g 0 B @ 1 2 1 2 v u x + u x y ` w x + u z 1 2 1 2 u + v y x w y v y + v z ` 1 u 2 v 1 2 z + w x + w z y w z 1 C A = A(T )σn 1 eff 0 @ σ xx σ xy σ xz σ yx σ yy σ yz σ zx σ zy σ zz 1 A Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 20 / 24

Mass equation (incompressibility) Temperature Equation u x + v y + w z = 0 ( T ρc p t + u T x + v T y + w T ) ( 2 ) T = k z x 2 + 2 T y 2 + 2 T z 2 + Φ Thickness equation H t = s B u dz + M s M B = (uh) + ȧ ṁ B = H ρ i ρ o Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 21 / 24

Momentum Equation Start with Navier-Stokes x-equation: After depth integration: u t + (uu, uv, uw) fv = 1 ρ (p x + i σ xi ) (DU) t + (DUU, DUV ) DfV = (K h D U)+ gd 2 2ρ 0 ρ x + g DA x c d U (U, V ) D = B(x, y, t) A(x, y, t) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 22 / 24

Thickness equation (from incompressibility) Temperature equation D t + (DU, DV ) = ṁ + ė (DT ) t + (DUT, DVT ) = (K h D T )+ Salinity equation T A ė + T B ṁ γ T (U, V ) (T T B ) (DS) t + (DUS, DVS) = (K h D S)+ S A ė + S B ṁ γ S (U, V ) (S S B ) Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 23 / 24

Plume thermodynamics Linearized phase-transition boundary: Linearized equation of state: T B = as B + b cb ρ = ρ 0 (1 + β S (S S 0 ) β T (T T 0 )). Melting is given by the flux balances: γ T (U, V ) (T T B ) = ṁl + ṁc ice (T B T I ) γ S (U, V ) (S S B ) = ṁs B Carl Gladish (NYU CIMS) Glimmer-CISM + Plume February 17, 2010 24 / 24