You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
<p>therefore has a similar interpretation as in PCA:
172
172
the first principal component (eigenvector <spanclass="arithmatex">\({\bf m}_1\)</span>) is the direction that maximizes the variance (eigenvalue <spanclass="arithmatex">\(\lambda_1\)</span>) of the projected data (red curve), the second component is the direction orthogonal to the first component that maximizes the variance of the projected data, and so on with the third component. </p>
<p>If ice grains are treated as transversely isotropic, the rheology of a single grain can be modeled as a <ahref="../constitutive-viscoplastic/">transversely isotropic power law</a>.
144
147
This requires specifying the grain eigenenhancements <spanclass="arithmatex">\(E_{cc}'\)</span> and <spanclass="arithmatex">\(E_{ca}'\)</span> and the power law exponent <spanclass="arithmatex">\(n'\)</span>. </p>
148
+
</div>
149
+
<divstyle="clear: both;"></div>
145
150
<p>The grain parameters proposed by <ahref="https://doi.org/10.1017/jog.2021.88">Rathmann and Lilien (2021)</a> assume linear-viscous behavior of single crystals (<spanclass="arithmatex">\(n'=1\)</span>) and promote the activation of basal glide by making that slip system soft compared to other systems: <spanclass="arithmatex">\(E_{ca}' > 1\)</span>, whereas <spanclass="arithmatex">\(E_{cc}'=1\)</span>.
146
151
This reduces the problem to picking <spanclass="arithmatex">\(E_{ca}'\)</span> and <spanclass="arithmatex">\(\alpha\)</span> (Taylor—Sachs homogenization weight), which <ahref="https://doi.org/10.1017/jog.2021.88">Rathmann and Lilien (2021)</a> determined by requiring that deformation tests on strong single-maximum CPOs (aligned grains) are approximately reproduced; that is, <spanclass="arithmatex">\(E_{mt}=10\)</span> and <spanclass="arithmatex">\(E_{mm}=0.01\)</span>.</p>
147
152
<p>The effect of choosing different <spanclass="arithmatex">\(E_{ca}'\)</span> and <spanclass="arithmatex">\(\alpha\)</span> (left panel) on the eigenenhancements of different CPO states (right panel) is shown below for combinations of <spanclass="arithmatex">\(E_{ca}'\)</span> and <spanclass="arithmatex">\(\alpha\)</span> that fulfill <spanclass="arithmatex">\(E_{mt}=10\)</span> given a unidirectional CPO. </p>
<p>If grains are approximately orthotropic, the grain rheology can be modelled using the <ahref="../constitutive-viscoplastic/">orthotropic power-law rheology</a>.
142
145
This requires specifying the grain eigenenhancements <spanclass="arithmatex">\(E_{ij}'\)</span>, the power-law exponent <spanclass="arithmatex">\(n'\)</span>, and the Taylor—Sachs weight <spanclass="arithmatex">\(\alpha\)</span>.</p>
146
+
</div>
147
+
<divstyle="clear: both;"></div>
143
148
<p>🚧 <em>Not yet documented — see <ahref="https://doi.org/10.1029/2024GC011831">Rathmann et al. (2024)</a> for details.</em></p>
<p>Polygonization (rotation recrystallization, CDRX) accounts for the division of grains along internal sub-grain boundaries resulting from local strain incompatibilities.
146
149
In effect, CDRX reduces the average grain size upon grain division but does not necessarily change the CPO much (<ahref="https://doi.org/10.3189/S0022143000003658">Alley, 1992</a>). </p>
150
+
</div>
151
+
<divstyle="clear: both;"></div>
147
152
<p>Following <ahref="https://doi.org/10.1007/s001610050095">Gödert (2003)</a>, CDRX can be modeled by approximating this effect as a Laplacian diffusive process on <spanclass="arithmatex">\(S^2\)</span>:</p>
148
153
<divclass="arithmatex">\[
149
154
\frac{\mathrm{D} n}{\mathrm{D} t} = \Lambda\nabla^2 n ,
<p>Following <ahref="https://doi.org/10.1007/s00161-009-0126-0">Placidi and others (2010)</a>, DDRX is modeled as a spontaneous mass decay—production process in orientation space <spanclass="arithmatex">\(S^2\)</span>, intended to represent the combined effect of nucleation and grain boundary migration.
149
+
That is, mass is spontaneously exchanged between grains with different orientations depending on the local stress state, strain rate, and temperature, in a statistical sense. </p>
150
+
</div>
151
+
<divstyle="clear: both;"></div>
145
152
<p>Following <ahref="https://doi.org/10.1007/s00161-009-0126-0">Placidi and others (2010)</a>, DDRX is modeled as a spontaneous mass decay—production process in orientation space <spanclass="arithmatex">\(S^2\)</span>, intended to represent the combined effect of nucleation and grain boundary migration.
146
153
That is, mass is spontaneously exchanged between grains with different orientations depending on the local stress state, strain rate, and temperature, in a statistical sense. </p>
147
154
<p>The decay—production rate is defined as</p>
<p>where <spanclass="arithmatex">\({\bf \bar s}(x,y)\)</span> is the depth-average CPO state vector field, <spanclass="arithmatex">\({\bf{u}}(x,y)=[u_x(x,y),u_y(x,y)]\)</span> is the horizontal surface velocity field, and <spanclass="arithmatex">\(H\)</span> is the ice thickness.
158
158
The first term represents CPO advection along stream lines, and the second term represents the depth-average effect of crystal processes.
159
159
The third and fourth terms are state-space attractors, causing <spanclass="arithmatex">\({\bf \bar s}\)</span> to tend towards the characteristic CPO states of ice that accumulates on the surface <spanclass="arithmatex">\({\bf s}_{\mathrm{sfc}}\)</span> or subglacially <spanclass="arithmatex">\({\bf s}_{\mathrm{sub}}\)</span>, depending on the positively-defined ice-equivalent accumulation rates <spanclass="arithmatex">\(a_{\mathrm{sfc}}\)</span> and <spanclass="arithmatex">\(a_{\mathrm{sub}}\)</span>. </p>
<p>If CPO development is dominated by lattice rotation (typical for cold ice), the problem is closed by specifying the horizontal surface velocity field (e.g., satellite-derived velocities), together with accumulation rates and the characteristic CPO state of accumulated ice (typically isotropic). </p>
163
163
<p>If DDRX is non-negligible (typical for warm ice), the temperature and stress field must additionally be prescribed.
strain = np.array([sf.F_to_strain(Fi[nn]) for nn in np.arange(Nt)]) # strain tensor
209
209
</code></pre>
210
210
<hr/>
211
-
<h2id="ice-core-cpos">Ice core CPOs</h2>
211
+
<h2id="ice-divide">Ice divide</h2>
212
212
<divstyle="float: left; width: 60%;">
213
213
<p>A Lagrangian approach is well-suited for modelling the vertical CPO profile at ice sheet domes and divides.
214
214
Assuming e.g. the classical Nye model of an ice divide of height <spanclass="arithmatex">\(H\)</span> (no basal melt, constant rate of thinning, a constant accumulation rate <spanclass="arithmatex">\(a\)</span>), the <ahref="../deformation-kinematics/">velocity gradient</a> is constant and equal to </p>
<p>If the Shallow Shelf/Stream Approximation (SSA) is applicable, velocities can be assumed depth constant (no vertical shearing).
379
382
In this case, a <ahref="../gallery-Eulerian-CPO-field/">depth-average treatment of CPO evolution</a> transforms the Lagrangian parcel model into a Lagrangian <em>column</em> model.
380
383
This generalizes the above parcel model, since the velocity gradient, stress and temperature fields can no longer be assumed constant but depend on the column position <spanclass="arithmatex">\({\bf x}(t)=[x(t),y(t)]\)</span>.</p>
0 commit comments