Docs for dynamics, switch to Glen's A

docs/centerlines.rst

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+.. _centerlines:
+
+Centerlines
+===========

docs/examples.rst

@@ -1,7 +1,7 @@
-Examples
-========
-
-Test for Latex:
-
-$$ a = b^2 + c^2$$
+Getting started
+===============
 
+The best way to get you started with OGGM is to run the example case study
+in the Öztal region. You will find a jupyter notebook
+[here](https://github.com/OGGM/oggm/blob/master/docs/notebooks/getting_started.ipynb)
+or in the ``oggm/docs/notebooks`` directory.

docs/flowline.rst

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+Ice dynamics
+============
+
+The glaciers in OGGM are represented by a depth integrated flowline
+model. The equations for the isothermal shallow ice are solved along
+the glacier centerline, computed to represent best the flow of ice
+along the glacier (see for example `antarcticglaciers.org`_ for a general
+introduction about the various type of glacier models).
+
+.. _antarcticglaciers.org: http://www.antarcticglaciers.org/glaciers-and-climate/numerical-ice-sheet-models/hierarchy-ice-sheet-models-introduction/
+
+Here we present the basic physics and numerics of the two models
+implemented currently in OGGM, the ``FluxBasedModel`` (homegrown model with a
+rather simple numerical solver) and the ``MUSCLSuperBeeModel`` (mass-conserving
+numerical scheme, see [Jarosch_etal_2013]_).
+
+
+
+Basics
+------
+
+Let :math:`S` be the area of a cross-section perpendicular to the
+flowline. It has a width :math:`w` and a thickness :math:`h` and, in this
+example, a parabolic bed shape.
+
+    .. figure:: ../files/hef_flowline.jpg
+        :width: 80%
+
+        Example of a cross-section along the glacier flowline. Background
+        image from
+        http://www.swisseduc.ch/glaciers/alps/hintereisferner/index-de.html
+
+Volume conservation for this discrete element implies:
+
+.. math::
+
+    \frac{\partial S}{\partial t} = w \, \dot{m} - \nabla \cdot q
+
+where :math:`\dot{m}` is the mass-balance, :math:`q = u S` the flux of ice, and
+:math:`u` the depth-integrated ice velocity ([Cuffey_Paterson_2010]_, p 310).
+This velocity can be computed from Glen's flow law as a function of the
+basal shear stress :math:`\tau`:
+
+.. math::
+
+    u = u_d + u_s = f_d h \tau^n + f_s \frac{\tau^n}{h}
+
+The second term is to account for basal sliding, see e.g. [Oerlemans_1997]_ or
+[Golledge_Levy_2011]_. It introduces an additional free parameter :math:`f_s`
+and will therefore be ignored in a first approach. The deformation parameter
+:math:`f_d` is better constrained and relates to Glen's
+temperature‐dependent creep parameter :math:`A`:
+
+.. math::
+
+    f_d = \frac{2 A}{n + 2}
+
+The basal shear stress :math:`\tau` depends e.g. on the geometry of the bed
+[Cuffey_Paterson_2010]_. Currently it is assumed to be
+equal to the driving stress :math:`\tau_d`:
+
+.. math::
+
+    \tau_d = \alpha \rho g h
+
+where :math:`\alpha` is the slope of the flowline and :math:`\rho` the density
+of ice. Both the ``FluxBasedModel`` and the ``MUSCLSuperBeeModel`` solve
+for these equations, but with different numerical schemes.
+
+Bed thickness inversion
+-----------------------
+
+To compute the initial ice thikness :math:`h_0`, OGGM follows a methodology
+largely inspired from
+[Farinotti_etal_2009]_ but using a different apparent mass-balance
+(see also: :ref:`mass-balance`) and another calibration algorithm.
+
+The principle is simple. Let's assume for now that we know the ice velocity
+:math:`u` along the flowline of our present-time glacier. Then the
+above equations can be used to compute the section area :math:`S` out of
+:math:`u` and the other ice-flow parameters. Since we know the present-time
+width :math:`w` with accuracy, :math:`h_0` can be obtained by assuming a
+certain geometrical shape for the bed.
+
+In OGGM, a number of climate and glacier related parameters are fixed prior to
+the inversion, leaving only one free parameter for the calibration of the
+bed inversion procedure: the inversion factor :math:`f_{inv}`. It is defined
+such as:
+
+.. math::
+
+    A = f_{inv} \, A_0
+
+With :math:`A_0` the standard creep parameter (2.4e-24). Currently,
+:math:`f_{inv}` is calibrated to minimize the volume RMSD of all glaciers
+with a volume estimation in the `GlaThiDa`_ database. It is therefore
+neither glacier nor temperature dependent and does not account for
+uncertainties in GlaThiDa's glacier-wide thickness estimations, two
+approximations which should be better handled in the future.
+
+.. _parabolic shape: https://en.wikipedia.org/wiki/Parabola#Area_enclosed_between_a_parabola_and_a_chord
+
+.. _GlaThiDa: http://www.gtn-g.ch/data_catalogue_glathida/
+
+
+Flux based model
+----------------
+
+Most flowline models treat the volume conservation equation as a
+diffusion problem, taking advantage of the robust numerical solutions
+available for this type of equations. The problem with this approach is that
+it develops the :math:`\partial S / \partial t` term to solve for
+ice thikness :math:`h` directly, thus implying different diffusion equations
+for different bed geometries (e.g. [Oerlemans_1997]_ with a trapezoidal bed).
+
+The OGGM flux based model solves for the :math:`\nabla \cdot q` term on a
+staggered grid (hence the name). It has the advantage that the model numerics
+are the same for any bed shape, but it makes one important simplification:
+the stress :math:`\tau = \alpha \rho g h` is always the same, regardless of the
+bed shape. This doesn't mean that the shape has no influence on the
+glacier evolution, as explained below.
+
+
+Glacier bed shapes
+------------------
+
+OGGM implements a number of possible bed-shapes. Currently the shape has no
+direct influence on ice dynamics, but it does influence how the width of the
+glacier changes with ice thickness and thus will influence the mass-balance
+:math:`w \, \dot{m}`. It appears that the flowline model is quite sensitive
+to the bed shape.
+
+
+VerticalWallFlowline
+~~~~~~~~~~~~~~~~~~~~
+
+
+    .. figure:: ../files/bed_vertical.png
+        :width: 40%
+
+
+The simplest shape. The glacier width does not change with ice thickness.
+
+
+TrapezoidalFlowline
+~~~~~~~~~~~~~~~~~~~
+
+
+    .. figure:: ../files/bed_trapezoidal.png
+        :width: 40%
+
+
+Trapezoidal shape with two degrees of freedom. The width change with thickness
+depends on :math:`\lambda`. [Golledge_Levy_2011]_ uses :math:`\lambda = 1`
+(a 45° wall angle).
+
+
+ParabolicFlowline
+~~~~~~~~~~~~~~~~~
+
+
+    .. figure:: ../files/bed_parabolic.png
+        :width: 40%
+
+
+Parabolic shape with one degree of freedom, which makes it particulary
+useful for the bed inversion: if :math:`S` and :math:`w` are known:
+
+.. math::
+
+    h = \frac{3}{2} \frac{S}{w}
+
+The parabola is defined by the single bed-shape parameter
+:math:`P_s = 4 h / w^2`. Very small values of this parameter imply very
+`flat` shapes, unrealistically sensitive to changes in :math:`h`. For this
+reason, the default in OGGM is to use the mixed flowline model.
+
+MixedFlowline
+~~~~~~~~~~~~~
+
+A combination of trapezoidal and parabolic flowlines. If the bed shape
+parameter :math:`P_s` is below a certain threshold, a trapezoidal shape is
+used instead.
+
+
+MUSCLSuperBeeModel
+------------------
+
+A shallow ice model with improved numerics ensuring mass-conservation in
+very steep walls [Jarosch_etal_2013]_. The model is currently in
+development to account for various bed shapes and tributaries and will
+likely become the default in OGGM.
+
+
+Glacier tributaries
+-------------------
+
+Glaciers in OGGM have a main centerline and, sometimes, one or more
+tributaries (which can themsleves also have tributaries, see
+:ref:`centerlines`). The number of these tributaries depends on many
+factors, but most of the time the algorithm works well.
+
+The main flowline and its tributaries are all handled the same way and
+are modelled individually. The difference is that tributaries can transport
+mass to the branch they are flowing to. Numerically, this mass transport is
+handled by adding an element at the end of the flowline with the same
+properties (with, thickness...) as the last grid point, with the difference
+that the slope :math:`\alpha` is computed with respect to the altitude of
+the point they are flowing to. The ice flux is then computed normally and
+transferred to the downlying branch.
+
+The computation of the ice flux is always done first from the lowest order
+branches (without tributaries) to the highest ones, ensuring a correct
+mass-redistribution. The angle between tributary and main branch ensures
+that the former is not decoupled from the latter. If the angle is positive
+or if no ice is present at the end of the tributary, no mass exchange occurs.
+
+
+References
+----------
+
+.. [Cuffey_Paterson_2010] Cuffey, K., and W. S. B. Paterson (2010).
+    The Physics of Glaciers, Butterworth‐Heinemann, Oxford, U.K.
+
+.. [Farinotti_etal_2009] Farinotti, D., Huss, M., Bauder, A., Funk, M., &
+    Truffer, M. (2009). A method to estimate the ice volume and
+    ice-thickness distribution of alpine glaciers. Journal of Glaciology, 55
+    (191), 422–430.
+
+.. [Golledge_Levy_2011] Golledge, N. R., and Levy, R. H. (2011).
+    Geometry and dynamics of an East Antarctic Ice Sheet outlet glacier, under
+    past and present climates. Journal of Geophysical Research:
+    Earth Surface, 116(3), 1–13.
+
+.. [Jarosch_etal_2013] Jarosch, a. H., Schoof, C. G., & Anslow, F. S. (2013).
+    Restoring mass conservation to shallow ice flow models over complex
+    terrain. Cryosphere, 7(1), 229–240. http://doi.org/10.5194/tc-7-229-2013
+
+.. [Oerlemans_1997] Oerlemans, J. (1997).
+    A flowline model for Nigardsbreen, Norway:
+    projection of future glacier length based on dynamic calibration with the
+    historic record. Journal of Glaciology, 24, 382–389.

docs/index.rst

@@ -27,7 +27,10 @@ Documentation
 
     examples
     installing-oggm
-    glacierdir-gen.rst
+    centerlines
+    mass-balance
+    flowline
+    glacierdir-gen
     api
 
 

docs/mass-balance.rst

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+.. _mass-balance:
+
+Mass-balance
+============