GlaDS Hydrology model

Description

The two-dimensional Glacier Drainage System model (GlaDS, [Werder2013]) couples a distributed water sheet model - a continuum description of a linked cavity drainage system [Hewitt2011] - with a channelized water flow model - modeled as R channels [Rothlisberger1972,Nye1976]. The coupled system collectively describes the evolution of hydraulic potential LaTeX equation, water sheet thickness LaTeX equation, and water channel cross-sectional area LaTeX equation.

Sheet model equations

  • Mass conservation: The mass conservation equation describes water storage changes over longer timescales (dictated by cavity opening due to sliding) as well as shorter timescales (e.g. due to surface melt water input):
    LaTeX equation
    where: LaTeX equation is the englacial void ratio, LaTeX equation is water density (kg m-3), LaTeX equation is gravitational acceleration (kg m-3), LaTeX equation is the hydraulic potential (Pa), and LaTeX equation is the sheet thickness (m). The water discharge LaTeX equation (m2 s-1) is given by:
    LaTeX equation
    where LaTeX equation is the sheet conductivity (m s kg-1), and LaTeX equationLaTeX equation5/4 and LaTeX equationLaTeX equation3/2 are two constant exponents. Finally, the melt source term LaTeX equation (m s-1) is given by:
    LaTeX equation
    where LaTeX equation is the geothermal heat flux (W m-2), LaTeX equation is the frictional heating (W m-2), for basal stress LaTeX equation (Pa), LaTeX equation is ice density (kg m-3), and LaTeX equation is latent heating (J kg-1).
  • Sheet thickness:
    LaTeX equation
    Here, LaTeX equation is the cavity opening rate due to sliding over bed topography (m s-1), given by:
    LaTeX equation
    where LaTeX equation and LaTeX equation are both constants (m), and LaTeX equation is the basal sliding velocity vector (provided by the ice flow model). The cavity closing rate due to ice creep LaTeX equation (m s-1), is given by:
    LaTeX equation
    where LaTeX equation is the basal flow parameter (Pa-3 s-1), related to the ice hardness by LaTeX equation-1/3, LaTeX equation is the Glen flow relation exponent, and LaTeX equation is the effective pressure. The overburden hydraulic potential is given by LaTeX equation, with the ice pressure LaTeX equation and elevation potential LaTeX equation, all of which are given in units of Pa.

Channel model equations

  • Channel discharge (along mesh edges):
    LaTeX equation
    where LaTeX equation is the channel discharge (m3 s-1), which evolves with respect to the horizontal coordinate along the channel LaTeX equation, LaTeX equation is the channel potential energy dissipated per unit length and time (W m-1), LaTeX equation is the channel pressure melting/refreezing (W m-1), LaTeX equation is the channel closing rate (m2 s-1) and LaTeX equation is the source term (m2 s-1). The discharge LaTeX equation is defined as:
    LaTeX equation
    where LaTeX equation is the channel conductivity, and LaTeX equationLaTeX equation3 and LaTeX equationLaTeX equation2 are two exponents. The term LaTeX equation is the closing of the channels by ice creep, and is given by:
    LaTeX equation
    where LaTeX equation is the channel cross-sectional area (m2). Finally, LaTeX equation, the channel source term balancing the flow of water out of the adjacent sheet, is:
    LaTeX equation
    where LaTeX equation is the normal to the channel edge.
  • Channel dissipation of potential energy:
    LaTeX equation
    where LaTeX equation is the channel width (m), and LaTeX equation is the discharge in the sheet (flowing in the direction of the channel; m2 s-1), given by:
    LaTeX equation
    with LaTeX equation, LaTeX equation, and LaTeX equation as given above.
  • Channel melt/refreeze rate:
    LaTeX equation
    Here, LaTeX equation is the Clapeyron slope (K Pa-1), LaTeX equation is the specific heat capacity of water (J kg-1 K-1), and LaTeX equation is a switch parameter that accounts for the fact that the channel area cannot be negative, turning off the sheet flow for refreezing as LaTeX equation, i.e.:
    LaTeX equation
  • Cross-sectional channel area (defined along mesh edges):
    LaTeX equation

Boundary conditions

Boundary conditions for the evolution of hydraulic potential LaTeX equation are applied on the domain boundary LaTeX equation, as either a prescribed pressure or water flux. The Dirichlet boundary condition is:

LaTeX equation

where LaTeX equation is a specific potential, and the Neumann boundary condition is:

LaTeX equation

corresponding to the specific discharge

LaTeX equation

Channels are defined only on element edges and are not allowed to cross the domain boundary, so we do not require flux conditions. Since the evolution equations for LaTeX equation and LaTeX equation do not contain their spatial derivatives, we do not require any boundary conditions for their evolution equations.

Model parameters

The parameters relevant to the GlaDS (hydrologyglads) solution can be displayed by running:

>> md.hydrology
  • md.hydrology.pressure_melt_coefficient: Pressure melt coefficient (LaTeX equation) [K Pa-1]
  • md.hydrology.sheet_conductivity: sheet conductivity (LaTeX equation) [m7/4 kg-1/2]
  • md.hydrology.cavity_spacing: cavity spacing (LaTeX equation) [m]
  • md.hydrology.bump_height: typical bump height (LaTeX equation) [m]
  • md.hydrology.ischannels: Do we allow for channels? 1: yes, 0: no
  • md.hydrology.channel_conductivity: channel conductivity (LaTeX equation) [m3/2 kg-1/2]
  • md.hydrology.spcphi: Hydraulic potential Dirichlet constraints [Pa]
  • md.hydrology.neumannflux: water flux applied along the model boundary (m2 s-1)
  • md.hydrology.moulin_input: moulin input (LaTeX equation) [m3 s-1]
  • md.hydrology.englacial_void_ratio: englacial void ratio (LaTeX equation)
  • md.hydrology.requested_outputs: additional outputs requested?
  • md.hydrology.melt_flag: User specified basal melt? 0: no (default), 1: use md.basalforcings.groundedice_melting_rate

Running a simulation

To run a transient standalone subglacial hydrology simulation, use the following commands:

md.transient=deactivateall(md.transient); md.transient.ishydrology=1; %Change hydrology class to GlaDS; md.hydrology=hydrologyglads(); %Set model parameters here; md=solve(md,'Transient');

References

  • Ian J. Hewitt. Modelling distributed and channelized subglacial drainage: the spacing of channels. J. Glaciol., 57(202):302-314, 2011.
  • J. F. Nye. Water flow in glaciers: jokulhlaups, tunnels and veins. J. Glaciol., 17(76):181-207, 1976.
  • H. Rothlisberger. Water pressure in intra-and subglacial channels. J. Glaciol., 11(62):177-203, 1972.
  • Mauro A. Werder, Ian J. Hewitt, Christian G. Schoof, and Gwenn E. Flowers. Modeling channelized and distributed subglacial drainage in two dimensions. J. Geophys. Res., 118:1-19, 2013.