| 592 |
$E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly |
$E_{0}\Delta{t}=\mbox{forcing time scale}$, or directly |
| 593 |
\code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. |
\code{SEAICE\_evpTauRelax} ($T$) to the forcing time scale. |
| 594 |
|
|
| 595 |
|
\paragraph{More stable variant of Elastic-Viscous-Plastic Dynamics: EVP*\label{sec:pkg:seaice:EVPstar}}~\\ |
| 596 |
|
% |
| 597 |
|
The genuine EVP schemes appears to give noisy solutions \citep{hun01, |
| 598 |
|
lemieux12, bouillon13}. This has lead to a modified EVP or EVP* |
| 599 |
|
\citep{lemieux12, bouillon13, kimmritz15}; here, refer to these |
| 600 |
|
variants by EVP*. The main idea is to modify the ``natural'' |
| 601 |
|
time-discretization of the momentum equations: |
| 602 |
|
\begin{equation} |
| 603 |
|
\label{eq:evpstar} |
| 604 |
|
m\frac{D\vec{u}}{Dt} \approx m\frac{u^{p+1}-u^{n}}{\Delta{t}} |
| 605 |
|
+ \beta^{*}\frac{u^{p+1}-u^{p}}{\Delta{t}_{\mathrm{EVP}}} |
| 606 |
|
\end{equation} |
| 607 |
|
where $n$ is the previous time step index, and $p$ is the previous |
| 608 |
|
sub-cycling index. The term allows the definition of a residual |
| 609 |
|
$|u^{p+1}-u^{p}|$ that, as $u^{p+1} \rightarrow u^{n+1}$, converges to |
| 610 |
|
$0$ and a re-interpretation of EVP as a pure iterative solver where |
| 611 |
|
the sub-cycling has lost all time-relation \citep{bouillon13, |
| 612 |
|
kimmritz15}. Using the terminology of \citet{kimmritz15}, the |
| 613 |
|
evolution equations of stress $\sigma_{ij}$ and momentum $\vec{u}$ can |
| 614 |
|
be written as: |
| 615 |
|
\begin{align} |
| 616 |
|
\label{eq:evpstarsigma} |
| 617 |
|
\sigma_{ij}^{p+1}&=\sigma_{ij}^p+\frac{1}{\alpha} |
| 618 |
|
\Big(\sigma_{ij}(\vec{u}^p)-\sigma_{ij}^p\Big), |
| 619 |
|
\phantom{\int}\\ |
| 620 |
|
\label{eq:evpstarmom} |
| 621 |
|
\vec{u}^{p+1}&=\vec{u}^p+\frac{1}{\beta} |
| 622 |
|
\Big(\frac{\Delta t}{m}\nabla \cdot{\bf \sigma}^{p+1}+ |
| 623 |
|
\frac{\Delta t}{m}\vec{R}^{p+1/2}+\vec{u}_n-\vec{u}^p\Big). |
| 624 |
|
\end{align} |
| 625 |
|
$\vec{R}$ contains all terms in the momentum equations except for the |
| 626 |
|
rheology terms and the time derivative, $\alpha$ and $\beta$ are free |
| 627 |
|
parameters (\code{SEAICE\_evpAlpha}, \code{SEAICE\_evpBeta}) that |
| 628 |
|
replace the time stepping parameters \code{SEAICE\_deltaTevp} |
| 629 |
|
($\Delta{T}_{\mathrm{EVP}}$), \code{SEAICE\_elasticParm} ($E_{0}$), or |
| 630 |
|
\code{SEAICE\_evpTauRelax} ($T$). $\alpha$ and $\beta$ determine the |
| 631 |
|
speed of convergence and the stability. Usually, it makes sense to use |
| 632 |
|
$\alpha = \beta$, and \code{SEAICEnEVPstarSteps} $>> \alpha = \beta$ |
| 633 |
|
\citep{kimmritz15}. |
| 634 |
|
|
| 635 |
|
In order to use EVP* in the MITgcm, set \code{SEAICEuseEVPstar = |
| 636 |
|
.TRUE.,} in \code{data.seaice}. \code{SEAICEuseEVPrev =.TRUE.,} uses |
| 637 |
|
the actual form of equations (\ref{eq:evpstarsigma}) and |
| 638 |
|
(\ref{eq:evpstarmom}) with fewer implicit terms and the factor of |
| 639 |
|
$e^{2}$ dropped in the stress equations (\ref{eq:evpstresstensor2}) |
| 640 |
|
and (\ref{eq:evpstresstensor12}). This turns out to improve |
| 641 |
|
convergence \citep{bouillon13}. |
| 642 |
|
|
| 643 |
|
Note, that for historical reasons, \code{SEAICE\_deltaTevp} needs to |
| 644 |
|
be set to some value in order to use also EVP*. Also note, that |
| 645 |
|
probably because of the C-grid staggering of velocities and stresses, |
| 646 |
|
EVP* does not converge as successfully as in \citet{kimmritz15}. |
| 647 |
|
|
| 648 |
|
|
| 649 |
\paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ |
\paragraph{Truncated ellipse method (TEM) for yield curve \label{sec:pkg:seaice:TEM}}~\\ |
| 650 |
% |
% |
| 651 |
In the so-called truncated ellipse method the shear viscosity $\eta$ |
In the so-called truncated ellipse method the shear viscosity $\eta$ |
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