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Update the calculation of uvel and vvel in evp dynamics #953
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@@ -35,11 +35,11 @@ For clarity, the two components of Equation :eq:`vpmom` are | |
\begin{aligned} | ||
m{\partial u\over\partial t} &= {\partial\sigma_{1j}\over\partial x_j} + \tau_{ax} + | ||
a_i c_w \rho_w | ||
\left|{\bf U}_w - {\bf u}\right| \left[\left(U_w-u\right)\cos\theta - \left(V_w-v\right)\sin\theta\right] | ||
\left|{\bf U}_w - {\bf u}\right| \left[ + \left(U_w-u\right)\cos\theta \mp \left(V_w-v\right)\sin\theta\right] | ||
-C_bu +mfv - mg{\partial H_\circ\over\partial x}, \\ | ||
m{\partial v\over\partial t} &= {\partial\sigma_{2j}\over\partial x_j} + \tau_{ay} + | ||
a_i c_w \rho_w | ||
\left|{\bf U}_w - {\bf u}\right| \left[\left(U_w-u\right)\sin\theta + \left(V_w-v\right)\cos\theta\right] | ||
\left|{\bf U}_w - {\bf u}\right| \left[ \pm \left(U_w-u\right)\sin\theta + \left(V_w-v\right)\cos\theta\right] | ||
-C_bv-mfu - mg{\partial H_\circ\over\partial y}. \end{aligned} | ||
:label: momsys | ||
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@@ -121,38 +121,38 @@ variables used in the code. | |
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.. math:: | ||
\underbrace{\left({m\over\Delta t_e}+{\tt vrel} \cos\theta\ + C_b \right)}_{\tt cca} u^{k+1} | ||
- \underbrace{\left(mf+{\tt vrel}\sin\theta\right)}_{\tt ccb}v^{l} | ||
- \underbrace{\left(mf \pm {\tt vrel}\sin\theta\right)}_{\tt ccb}v^{l} | ||
= &\underbrace{{\partial\sigma_{1j}^{k+1}\over\partial x_j}}_{\tt strintx} | ||
+ \underbrace{\tau_{ax} - mg{\partial H_\circ\over\partial x} }_{\tt forcex} \\ | ||
&+ {\tt vrel}\underbrace{\left(U_w\cos\theta-V_w\sin\theta\right)}_{\tt waterx} + {m\over\Delta t_e}u^k, | ||
&+ {\tt vrel}\underbrace{\left(+U_w\cos\theta \mp V_w\sin\theta\right)}_{\tt waterx} + {m\over\Delta t_e}u^k, | ||
:label: umom | ||
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.. math:: | ||
\underbrace{\left(mf+{\tt vrel}\sin\theta\right)}_{\tt ccb} u^{l} | ||
\underbrace{\left(mf \pm {\tt vrel}\sin\theta\right)}_{\tt ccb} u^{l} | ||
+ \underbrace{\left({m\over\Delta t_e}+{\tt vrel} \cos\theta + C_b \right)}_{\tt cca}v^{k+1} | ||
= &\underbrace{{\partial\sigma_{2j}^{k+1}\over\partial x_j}}_{\tt strinty} | ||
+ \underbrace{\tau_{ay} - mg{\partial H_\circ\over\partial y} }_{\tt forcey} \\ | ||
&+ {\tt vrel}\underbrace{\left(U_w\sin\theta+V_w\cos\theta\right)}_{\tt watery} + {m\over\Delta t_e}v^k, | ||
&+ {\tt vrel}\underbrace{\left( \pm U_w\sin\theta+V_w\cos\theta\right)}_{\tt watery} + {m\over\Delta t_e}v^k, | ||
:label: vmom | ||
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where :math:`{\tt vrel}\ \cdot\ {\tt waterx(y)}= {\tt taux(y)}` and the definitions of :math:`u^{l}` and :math:`v^{l}` vary depending on the grid. | ||
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As :math:`u` and :math:`v` are collocated on the B grid, :math:`u^{l}` and :math:`v^{l}` are respectively :math:`u^{k+1}` and :math:`v^{k+1}` such that this system of equations can be solved as follows. Define | ||
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.. math:: | ||
\hat{u} = F_u + \tau_{ax} - mg{\partial H_\circ\over\partial x} + {\tt vrel} \left(U_w\cos\theta - V_w\sin\theta\right) + {m\over\Delta t_e}u^k | ||
\hat{u} = F_u + \tau_{ax} - mg{\partial H_\circ\over\partial x} + {\tt vrel} \left(+ U_w\cos\theta \mp V_w\sin\theta\right) + {m\over\Delta t_e}u^k | ||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. same comment here not sure I would add the + sign before U_w There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This was done just to keep the equations aligned a bit in the text. I'll remove them again, wasn't sure what was better. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. The extra plus signs are out, feel free to have another look. Can revert if you prefer the prior version. If I don't hear anything, will merge later today. |
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:label: cevpuhat | ||
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.. math:: | ||
\hat{v} = F_v + \tau_{ay} - mg{\partial H_\circ\over\partial y} + {\tt vrel} \left(U_w\sin\theta + V_w\cos\theta\right) + {m\over\Delta t_e}v^k, | ||
\hat{v} = F_v + \tau_{ay} - mg{\partial H_\circ\over\partial y} + {\tt vrel} \left(\pm U_w\sin\theta + V_w\cos\theta\right) + {m\over\Delta t_e}v^k, | ||
:label: cevpvhat | ||
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where :math:`{\bf F} = \nabla\cdot\sigma^{k+1}`. Then | ||
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.. math:: | ||
\begin{aligned} | ||
\left({m\over\Delta t_e} +{\tt vrel}\cos\theta\ + C_b \right)u^{k+1} - \left(mf + {\tt vrel}\sin\theta\right) v^{k+1} &= \hat{u} \\ | ||
\left(mf + {\tt vrel}\sin\theta\right) u^{k+1} + \left({m\over\Delta t_e} +{\tt vrel}\cos\theta + C_b \right)v^{k+1} &= \hat{v}.\end{aligned} | ||
\left({m\over\Delta t_e} +{\tt vrel}\cos\theta\ + C_b \right)u^{k+1} - \left(mf \pm {\tt vrel}\sin\theta\right) v^{k+1} &= \hat{u} \\ | ||
\left(mf \pm {\tt vrel}\sin\theta\right) u^{k+1} + \left({m\over\Delta t_e} +{\tt vrel}\cos\theta + C_b \right)v^{k+1} &= \hat{v}.\end{aligned} | ||
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Solving simultaneously for :math:`u^{k+1}` and :math:`v^{k+1}`, | ||
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@@ -168,7 +168,7 @@ where | |
:label: cevpa | ||
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.. math:: | ||
b = mf + {\tt vrel}\sin\theta. | ||
b = mf \pm {\tt vrel}\sin\theta. | ||
:label: cevpb | ||
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Note that the time discretization and solution method for the EAP is exactly the same as for the B grid EVP. More details on the EAP model are given in Section :ref:`stress-eap`. | ||
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@@ -191,39 +191,39 @@ implicit solvers and there is an additional term for the pseudo-time iteration. | |
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.. math:: | ||
{\beta^*(u^{k+1}-u^k)\over\Delta t_e} + {m(u^{k+1}-u^n)\over\Delta t} + {\left({\tt vrel} \cos\theta + C_b \right)} u^{k+1} | ||
- & {\left(mf+{\tt vrel}\sin\theta\right)} v^{l} | ||
- & {\left(mf \pm {\tt vrel}\sin\theta\right)} v^{l} | ||
= {{\partial\sigma_{1j}^{k+1}\over\partial x_j}} | ||
+ {\tau_{ax}} \\ | ||
& - {mg{\partial H_\circ\over\partial x} } | ||
+ {\tt vrel} {\left(U_w\cos\theta-V_w\sin\theta\right)}, | ||
+ {\tt vrel} {\left(+U_w\cos\theta \mp V_w\sin\theta\right)}, | ||
:label: umomr | ||
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.. math:: | ||
{\beta^*(v^{k+1}-v^k)\over\Delta t_e} + {m(v^{k+1}-v^n)\over\Delta t} + {\left({\tt vrel} \cos\theta + C_b \right)}v^{k+1} | ||
+ & {\left(mf+{\tt vrel}\sin\theta\right)} u^{l} | ||
+ & {\left(mf \pm {\tt vrel}\sin\theta\right)} u^{l} | ||
= {{\partial\sigma_{2j}^{k+1}\over\partial x_j}} | ||
+ {\tau_{ay}} \\ | ||
& - {mg{\partial H_\circ\over\partial y} } | ||
+ {\tt vrel}{\left(U_w\sin\theta+V_w\cos\theta\right)}, | ||
+ {\tt vrel}{\left( \pm U_w\sin\theta+V_w\cos\theta\right)}, | ||
:label: vmomr | ||
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where :math:`\beta^*` is a numerical parameter and :math:`u^n, v^n` are the components of the previous time level solution. | ||
With :math:`\beta=\beta^* \Delta t \left( m \Delta t_e \right)^{-1}` :cite:`Bouillon13`, these equations can be written as | ||
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.. math:: | ||
\underbrace{\left((\beta+1){m\over\Delta t}+{\tt vrel} \cos\theta\ + C_b \right)}_{\tt cca} u^{k+1} | ||
- \underbrace{\left(mf+{\tt vrel} \sin\theta\right)}_{\tt ccb} & v^{l} | ||
- \underbrace{\left(mf \pm {\tt vrel} \sin\theta\right)}_{\tt ccb} & v^{l} | ||
= \underbrace{{\partial\sigma_{1j}^{k+1}\over\partial x_j}}_{\tt strintx} | ||
+ \underbrace{\tau_{ax} - mg{\partial H_\circ\over\partial x} }_{\tt forcex} \\ | ||
& + {\tt vrel}\underbrace{\left(U_w\cos\theta-V_w\sin\theta\right)}_{\tt waterx} + {m\over\Delta t}(\beta u^k + u^n), | ||
& + {\tt vrel}\underbrace{\left(+U_w\cos\theta \mp V_w\sin\theta\right)}_{\tt waterx} + {m\over\Delta t}(\beta u^k + u^n), | ||
:label: umomr2 | ||
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.. math:: | ||
\underbrace{\left(mf+{\tt vrel}\sin\theta\right)}_{\tt ccb} u^{l} | ||
\underbrace{\left(mf \pm {\tt vrel}\sin\theta\right)}_{\tt ccb} u^{l} | ||
+ \underbrace{\left((\beta+1){m\over\Delta t}+{\tt vrel} \cos\theta + C_b \right)}_{\tt cca} & v^{k+1} | ||
= \underbrace{{\partial\sigma_{2j}^{k+1}\over\partial x_j}}_{\tt strinty} | ||
+ \underbrace{\tau_{ay} - mg{\partial H_\circ\over\partial y} }_{\tt forcey} \\ | ||
& + {\tt vrel}\underbrace{\left(U_w\sin\theta+V_w\cos\theta\right)}_{\tt watery} + {m\over\Delta t}(\beta v^k + v^n), | ||
& + {\tt vrel}\underbrace{\left( \pm U_w\sin\theta+V_w\cos\theta\right)}_{\tt watery} + {m\over\Delta t}(\beta v^k + v^n), | ||
:label: vmomr2 | ||
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At this point, the solutions :math:`u^{k+1}` and :math:`v^{k+1}` for the B or the C grids are obtained in the same manner as for the standard EVP approach (see Section :ref:`evp-momentum` for details). | ||
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@@ -292,6 +292,8 @@ Ice-Ocean stress | |
At the end of each (thermodynamic) time step, the | ||
ice–ocean stress must be constructed from :math:`{\tt taux(y)}` and the terms | ||
containing :math:`{\tt vrel}` on the left hand side of the equations. | ||
The water stress calculation has a hemispheric dependence on the sign of the | ||
:math:`\pm {\tt vrel}\sin\theta` term. | ||
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The Hibler-Bryan form for the ice-ocean stress :cite:`Hibler87` | ||
is included in **ice\_dyn\_shared.F90** but is currently commented out, | ||
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not sure I would add the + sign before \left(U_w...