| 13 |
forward prognostic variables while satisfying constraints imposed by |
forward prognostic variables while satisfying constraints imposed by |
| 14 |
diagnostic equations. |
diagnostic equations. |
| 15 |
|
|
| 16 |
Since the model comes in several flavours and formulation, it would be |
Since the model comes in several flavors and formulation, it would be |
| 17 |
confusing to present the model algorithm exactly as written into code |
confusing to present the model algorithm exactly as written into code |
| 18 |
along with all the switches and optional terms. Instead, we present |
along with all the switches and optional terms. Instead, we present |
| 19 |
the algorithm for each of the basic formulations which are: |
the algorithm for each of the basic formulations which are: |
| 37 |
\ref{sect:pressure-method-rigid-lid}. This algorithm is essentially |
\ref{sect:pressure-method-rigid-lid}. This algorithm is essentially |
| 38 |
unchanged, apart for some coefficients, when the rigid lid assumption |
unchanged, apart for some coefficients, when the rigid lid assumption |
| 39 |
is replaced with a linearized implicit free-surface, described in |
is replaced with a linearized implicit free-surface, described in |
| 40 |
section \ref{sect:pressure-method-linear-backward}. These two flavours |
section \ref{sect:pressure-method-linear-backward}. These two flavors |
| 41 |
of the pressure-method encompass all formulations of the model as it |
of the pressure-method encompass all formulations of the model as it |
| 42 |
exists today. The integration of explicit in time terms is out-lined |
exists today. The integration of explicit in time terms is out-lined |
| 43 |
in section \ref{sect:adams-bashforth} and put into the context of the |
in section \ref{sect:adams-bashforth} and put into the context of the |
| 61 |
algorithm. A prediction for the flow variables at time level $n+1$ is |
algorithm. A prediction for the flow variables at time level $n+1$ is |
| 62 |
made based only on the explicit terms, $G^{(n+^1/_2)}$, and denoted |
made based only on the explicit terms, $G^{(n+^1/_2)}$, and denoted |
| 63 |
$u^*$, $v^*$. Next, a pressure field is found such that $u^{n+1}$, |
$u^*$, $v^*$. Next, a pressure field is found such that $u^{n+1}$, |
| 64 |
$v^{n+1}$ will be non-divergent. Conceptually, the $*$ quantitites |
$v^{n+1}$ will be non-divergent. Conceptually, the $*$ quantities |
| 65 |
exist at time level $n+1$ but they are intermediate and only |
exist at time level $n+1$ but they are intermediate and only |
| 66 |
temporary.} |
temporary.} |
| 67 |
\label{fig:pressure-method-rigid-lid} |
\label{fig:pressure-method-rigid-lid} |
| 80 |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
\>\> CALC\_GRAD\_PHI\_SURF \` $\nabla \eta^{n+1}$ \\ |
| 81 |
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid}) |
\>\> CORRECTION\_STEP \` $u^{n+1}$,$v^{n+1}$ (\ref{eq:un+1-rigid-lid},\ref{eq:vn+1-rigid-lid}) |
| 82 |
\end{tabbing} \end{minipage} } \end{center} |
\end{tabbing} \end{minipage} } \end{center} |
| 83 |
\caption{Calling tree for the pressure method alogtihm} |
\caption{Calling tree for the pressure method algorihtm} |
| 84 |
\label{fig:call-tree-pressure-method} |
\label{fig:call-tree-pressure-method} |
| 85 |
\end{figure} |
\end{figure} |
| 86 |
|
|
| 102 |
\label{eq:rigid-lid-continuity} |
\label{eq:rigid-lid-continuity} |
| 103 |
\end{equation} |
\end{equation} |
| 104 |
Here, $H\widehat{u} = \int_H u dz$ is the depth integral of $u$, |
Here, $H\widehat{u} = \int_H u dz$ is the depth integral of $u$, |
| 105 |
similarly for $H\widehat{v}$. The rigid-lid approzimation sets $w=0$ |
similarly for $H\widehat{v}$. The rigid-lid approximation sets $w=0$ |
| 106 |
at the lid so that it does not move but allows a pressure to be |
at the lid so that it does not move but allows a pressure to be |
| 107 |
exerted on the fluid by the lid. The horizontal momentum equations and |
exerted on the fluid by the lid. The horizontal momentum equations and |
| 108 |
vertically integrated continuity equation are be discretized in time |
vertically integrated continuity equation are be discretized in time |
| 130 |
$G$. |
$G$. |
| 131 |
|
|
| 132 |
Substituting the two momentum equations into the depth integrated |
Substituting the two momentum equations into the depth integrated |
| 133 |
continuity equation eliminates $u^{n+1}$ and $v^{n+1}$ yeilding an |
continuity equation eliminates $u^{n+1}$ and $v^{n+1}$ yielding an |
| 134 |
elliptic equation for $\eta^{n+1}$. Equations |
elliptic equation for $\eta^{n+1}$. Equations |
| 135 |
\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and |
\ref{eq:discrete-time-u}, \ref{eq:discrete-time-v} and |
| 136 |
\ref{eq:discrete-time-cont-rigid-lid} can then be re-arranged as follows: |
\ref{eq:discrete-time-cont-rigid-lid} can then be re-arranged as follows: |
| 157 |
with the future flow field (time level $n+1$) but it could equally |
with the future flow field (time level $n+1$) but it could equally |
| 158 |
have been drawn as staggered in time with the flow. |
have been drawn as staggered in time with the flow. |
| 159 |
|
|
| 160 |
The correspondance to the code is as follows: |
The correspondence to the code is as follows: |
| 161 |
\begin{itemize} |
\begin{itemize} |
| 162 |
\item |
\item |
| 163 |
the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
the prognostic phase, equations \ref{eq:ustar-rigid-lid} and \ref{eq:vstar-rigid-lid}, |
| 165 |
{\em TIMESTEP.F} |
{\em TIMESTEP.F} |
| 166 |
\item |
\item |
| 167 |
the vertical integration, $H \widehat{u^*}$ and $H |
the vertical integration, $H \widehat{u^*}$ and $H |
| 168 |
\widehat{v^*}$, divergence and invertion of the elliptic operator in |
\widehat{v^*}$, divergence and inversion of the elliptic operator in |
| 169 |
equation \ref{eq:elliptic} is coded in {\em |
equation \ref{eq:elliptic} is coded in {\em |
| 170 |
SOLVE\_FOR\_PRESSURE.F} |
SOLVE\_FOR\_PRESSURE.F} |
| 171 |
\item |
\item |
| 250 |
the pressure method algorithm with a backward implicit, linearized |
the pressure method algorithm with a backward implicit, linearized |
| 251 |
free surface. The method is still formerly a pressure method because |
free surface. The method is still formerly a pressure method because |
| 252 |
in the limit of large $\Delta t$ the rigid-lid method is |
in the limit of large $\Delta t$ the rigid-lid method is |
| 253 |
reovered. However, the implicit treatment of the free-surface allows |
recovered. However, the implicit treatment of the free-surface allows |
| 254 |
the flow to be divergent and for the surface pressure/elevation to |
the flow to be divergent and for the surface pressure/elevation to |
| 255 |
respond on a finite time-scale (as opposed to instantly). To recovere |
respond on a finite time-scale (as opposed to instantly). To recover |
| 256 |
the rigid-lid formulation, we introduced a switch-like parameter, |
the rigid-lid formulation, we introduced a switch-like parameter, |
| 257 |
$\epsilon_{fs}$, which selects between the free-surface and rigid-lid; |
$\epsilon_{fs}$, which selects between the free-surface and rigid-lid; |
| 258 |
$\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$ |
$\epsilon_{fs}=1$ allows the free-surface to evolve; $\epsilon_{fs}=0$ |
| 331 |
|
|
| 332 |
Vertical diffusion and viscosity can be treated implicitly in time |
Vertical diffusion and viscosity can be treated implicitly in time |
| 333 |
using the backward method which is an intrinsic scheme. For tracers, |
using the backward method which is an intrinsic scheme. For tracers, |
| 334 |
the time discrretized equation is: |
the time discretized equation is: |
| 335 |
\begin{equation} |
\begin{equation} |
| 336 |
\tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = |
\tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = |
| 337 |
\tau^{n} + \Delta t G_\tau^{(n+1/2)} |
\tau^{n} + \Delta t G_\tau^{(n+1/2)} |
| 377 |
\caption{ |
\caption{ |
| 378 |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
| 379 |
phases of the algorithm. All prognostic variables are co-located in |
phases of the algorithm. All prognostic variables are co-located in |
| 380 |
time. Explicit tendancies are evaluated at time level $n$ as a |
time. Explicit tendencies are evaluated at time level $n$ as a |
| 381 |
function of the state at that time level (dotted arrow). The explicit |
function of the state at that time level (dotted arrow). The explicit |
| 382 |
tendancy from the previous time level, $n-1$, is used to extrapolate |
tendency from the previous time level, $n-1$, is used to extrapolate |
| 383 |
tendancies to $n+1/2$ (dashed arrow). This extrapolated tendancy |
tendencies to $n+1/2$ (dashed arrow). This extrapolated tendency |
| 384 |
allows variables to be stably integrated forward-in-time to render an |
allows variables to be stably integrated forward-in-time to render an |
| 385 |
estimate ($*$-variables) at the $n+1$ time level (solid |
estimate ($*$-variables) at the $n+1$ time level (solid |
| 386 |
arc-arrow). The operator ${\cal L}$ formed from implicit-in-time terms |
arc-arrow). The operator ${\cal L}$ formed from implicit-in-time terms |
| 417 |
\label{fig:call-tree-adams-bashforth-sync} |
\label{fig:call-tree-adams-bashforth-sync} |
| 418 |
\end{figure} |
\end{figure} |
| 419 |
|
|
| 420 |
The Adams-Bashforth extrapolation of explicit tendancies fits neatly |
The Adams-Bashforth extrapolation of explicit tendencies fits neatly |
| 421 |
into the pressure method algorithm when all state variables are |
into the pressure method algorithm when all state variables are |
| 422 |
co-locacted in time. Fig.~\ref{fig:adams-bashforth-sync} illustrates |
co-located in time. Fig.~\ref{fig:adams-bashforth-sync} illustrates |
| 423 |
the location of variables in time and the evolution of the algorithm |
the location of variables in time and the evolution of the algorithm |
| 424 |
with time. The algorithm can be represented by the sequential solution |
with time. The algorithm can be represented by the sequential solution |
| 425 |
of the follow equations: |
of the follow equations: |
| 453 |
\end{eqnarray} |
\end{eqnarray} |
| 454 |
Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of |
Fig.~\ref{fig:adams-bashforth-sync} illustrates the location of |
| 455 |
variables in time and evolution of the algorithm with time. The |
variables in time and evolution of the algorithm with time. The |
| 456 |
Adams-Bashforth extrapolation of the tracer tendancies is illustrated |
Adams-Bashforth extrapolation of the tracer tendencies is illustrated |
| 457 |
byt the dashed arrow, the prediction at $n+1$ is indicated by the |
by the dashed arrow, the prediction at $n+1$ is indicated by the |
| 458 |
solid arc. Inversion of the implicit terms, ${\cal |
solid arc. Inversion of the implicit terms, ${\cal |
| 459 |
L}^{-1}_{\theta,S}$, then yields the new tracer fields at $n+1$. All |
L}^{-1}_{\theta,S}$, then yields the new tracer fields at $n+1$. All |
| 460 |
these operations are carried out in subroutine {\em THERMODYNAMICS} an |
these operations are carried out in subroutine {\em THERMODYNAMICS} an |
| 480 |
\caption{ |
\caption{ |
| 481 |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
A schematic of the explicit Adams-Bashforth and implicit time-stepping |
| 482 |
phases of the algorithm but with staggering in time of thermodynamic |
phases of the algorithm but with staggering in time of thermodynamic |
| 483 |
variables with the flow. Explicit thermodynamics tendancies are |
variables with the flow. Explicit thermodynamics tendencies are |
| 484 |
evaluated at time level $n-1/2$ as a function of the thermodynamics |
evaluated at time level $n-1/2$ as a function of the thermodynamics |
| 485 |
state at that time level $n$ and flow at time $n$ (dotted arrow). The |
state at that time level $n$ and flow at time $n$ (dotted arrow). The |
| 486 |
explicit tendancy from the previous time level, $n-3/2$, is used to |
explicit tendency from the previous time level, $n-3/2$, is used to |
| 487 |
extrapolate tendancies to $n$ (dashed arrow). This extrapolated |
extrapolate tendencies to $n$ (dashed arrow). This extrapolated |
| 488 |
tendancy allows thermo-dynamics variables to be stably integrated |
tendency allows thermo-dynamics variables to be stably integrated |
| 489 |
forward-in-time to render an estimate ($*$-variables) at the $n+1/2$ |
forward-in-time to render an estimate ($*$-variables) at the $n+1/2$ |
| 490 |
time level (solid arc-arrow). The implicit-in-time operator ${\cal |
time level (solid arc-arrow). The implicit-in-time operator ${\cal |
| 491 |
L_{\theta,S}}$ is solved to yield the thermodynamic variables at time |
L_{\theta,S}}$ is solved to yield the thermodynamic variables at time |
| 502 |
circumstance, it is more efficient to stagger in time the |
circumstance, it is more efficient to stagger in time the |
| 503 |
thermodynamic variables with the flow |
thermodynamic variables with the flow |
| 504 |
variables. Fig.~\ref{fig:adams-bashforth-staggered} illustrates the |
variables. Fig.~\ref{fig:adams-bashforth-staggered} illustrates the |
| 505 |
staggering and algorith. The key difference between this and |
staggering and algorithm. The key difference between this and |
| 506 |
Fig.~\ref{fig:adams-bashforth-sync} is that the new thermodynamics |
Fig.~\ref{fig:adams-bashforth-sync} is that the new thermodynamics |
| 507 |
fields are used to compute the hydrostatic pressure at time level |
fields are used to compute the hydrostatic pressure at time level |
| 508 |
$n+1/2$. The essentially allows the gravity wave terms to leap-frog in |
$n+1/2$. The essentially allows the gravity wave terms to leap-frog in |
| 515 |
\phi_{hyd}^n = \int b(\theta^{n+1},S^{n+1}) dr |
\phi_{hyd}^n = \int b(\theta^{n+1},S^{n+1}) dr |
| 516 |
\end{displaymath} |
\end{displaymath} |
| 517 |
but the pressure gradient must also be taken out of the |
but the pressure gradient must also be taken out of the |
| 518 |
Adams-Bashforth extrapoltion. Also, retaining the integer time-levels, |
Adams-Bashforth extrapolation. Also, retaining the integer time-levels, |
| 519 |
$n$ and $n+1$, does not give a user the sense of where variables are |
$n$ and $n+1$, does not give a user the sense of where variables are |
| 520 |
located in time. Instead, we re-write the entire algorithm, |
located in time. Instead, we re-write the entire algorithm, |
| 521 |
\ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the |
\ref{eq:Gt-n-sync} to \ref{eq:v-n+1-sync}, annotating the |
| 554 |
{\em PARM01} of {\em data}. |
{\em PARM01} of {\em data}. |
| 555 |
|
|
| 556 |
The only difficulty with this approach is apparent in equation |
The only difficulty with this approach is apparent in equation |
| 557 |
$\ref{eq:Gt-n-staggered}$ and illustrated by the dotted arrow |
\ref{eq:Gt-n-staggered} and illustrated by the dotted arrow |
| 558 |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
connecting $u,v^n$ with $G_\theta^{n-1/2}$. The flow used to advect |
| 559 |
tracers around is not naturally located in time. This could be avoided |
tracers around is not naturally located in time. This could be avoided |
| 560 |
by applying the Adams-Bashforth extrapolation to the tracer field |
by applying the Adams-Bashforth extrapolation to the tracer field |
| 561 |
itself and advection that around but this is not yet available. We're |
itself and advecting that around but this approach is not yet |
| 562 |
not aware of any detrimental effect of this feature. The difficulty |
available. We're not aware of any detrimental effect of this |
| 563 |
lies mainly in interpretation of what time-level variables and terms |
feature. The difficulty lies mainly in interpretation of what |
| 564 |
correspond to. |
time-level variables and terms correspond to. |
| 565 |
|
|
| 566 |
|
|
| 567 |
\section{Non-hydrostatic formulation} |
\section{Non-hydrostatic formulation} |
| 568 |
\label{sect:non-hydrostatic} |
\label{sect:non-hydrostatic} |
| 569 |
|
|
| 570 |
[to be written...] |
The non-hydrostatic formulation re-introduces the full vertical |
| 571 |
|
momentum equation and requires the solution of a 3-D elliptic |
| 572 |
|
equations for non-hydrostatic pressure perturbation. We still |
| 573 |
|
intergrate vertically for the hydrostatic pressure and solve a 2-D |
| 574 |
|
elliptic equation for the surface pressure/elevation for this reduces |
| 575 |
|
the amount of work needed to solve for the non-hydrostatic pressure. |
| 576 |
|
|
| 577 |
|
The momentum equations are discretized in time as follows: |
| 578 |
|
\begin{eqnarray} |
| 579 |
|
\frac{1}{\Delta t} u^{n+1} + g \partial_x \eta^{n+1} + \partial_x \phi_{nh}^{n+1} |
| 580 |
|
& = & \frac{1}{\Delta t} u^{n} + G_u^{(n+1/2)} \label{eq:discrete-time-u-nh} \\ |
| 581 |
|
\frac{1}{\Delta t} v^{n+1} + g \partial_y \eta^{n+1} + \partial_y \phi_{nh}^{n+1} |
| 582 |
|
& = & \frac{1}{\Delta t} v^{n} + G_v^{(n+1/2)} \label{eq:discrete-time-v-nh} \\ |
| 583 |
|
\frac{1}{\Delta t} w^{n+1} + \partial_r \phi_{nh}^{n+1} |
| 584 |
|
& = & \frac{1}{\Delta t} w^{n} + G_w^{(n+1/2)} \label{eq:discrete-time-w-nh} \\ |
| 585 |
|
\end{eqnarray} |
| 586 |
|
which must satisfy the discrete-in-time depth integrated continuity, |
| 587 |
|
equation~\ref{eq:discrete-time-backward-free-surface} and the local continuity equation |
| 588 |
|
\begin{equation} |
| 589 |
|
\partial_x u^{n+1} + \partial_y v^{n+1} + \partial_r w^{n+1} = 0 |
| 590 |
|
\label{eq:non-divergence-nh} |
| 591 |
|
\end{equation} |
| 592 |
|
As before, the explicit predictions for momentum are consolidated as: |
| 593 |
|
\begin{eqnarray*} |
| 594 |
|
u^* & = & u^n + \Delta t G_u^{(n+1/2)} \\ |
| 595 |
|
v^* & = & v^n + \Delta t G_v^{(n+1/2)} \\ |
| 596 |
|
w^* & = & w^n + \Delta t G_w^{(n+1/2)} |
| 597 |
|
\end{eqnarray*} |
| 598 |
|
but this time we introduce an intermediate step by splitting the |
| 599 |
|
tendancy of the flow as follows: |
| 600 |
|
\begin{eqnarray} |
| 601 |
|
u^{n+1} = u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} |
| 602 |
|
& & |
| 603 |
|
u^{**} = u^{*} - \Delta t g \partial_x \eta^{n+1} \\ |
| 604 |
|
v^{n+1} = v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} |
| 605 |
|
& & |
| 606 |
|
v^{**} = v^{*} - \Delta t g \partial_y \eta^{n+1} |
| 607 |
|
\end{eqnarray} |
| 608 |
|
Substituting into the depth integrated continuity |
| 609 |
|
(equation~\ref{eq:discrete-time-backward-free-surface}) gives |
| 610 |
|
\begin{equation} |
| 611 |
|
\partial_x H \partial_x \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
| 612 |
|
+ |
| 613 |
|
\partial_y H \partial_y \left( g \eta^{n+1} + \widehat{\phi}_{nh}^{n+1} \right) |
| 614 |
|
- \frac{\epsilon_{fs}\eta^*}{\Delta t^2} |
| 615 |
|
= - \frac{\eta^*}{\Delta t^2} |
| 616 |
|
\end{equation} |
| 617 |
|
which is approximated by equation |
| 618 |
|
\ref{eq:elliptic-backward-free-surface} on the basis that i) |
| 619 |
|
$\phi_{nh}^{n+1}$ is not yet known and ii) $\nabla \widehat{\phi}_{nh} |
| 620 |
|
<< g \nabla \eta$. If \ref{eq:elliptic-backward-free-surface} is |
| 621 |
|
solved accurately then the implication is that $\widehat{\phi}_{nh} |
| 622 |
|
\approx 0$ so that thet non-hydrostatic pressure field does not drive |
| 623 |
|
barotropic motion. |
| 624 |
|
|
| 625 |
|
The flow must satisfy non-divergence |
| 626 |
|
(equation~\ref{eq:non-divergence-nh}) locally, as well as depth |
| 627 |
|
integrated, and this constraint is used to form a 3-D elliptic |
| 628 |
|
equations for $\phi_{nh}^{n+1}$: |
| 629 |
|
\begin{equation} |
| 630 |
|
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
| 631 |
|
\partial_{rr} \phi_{nh}^{n+1} = |
| 632 |
|
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} |
| 633 |
|
\end{equation} |
| 634 |
|
|
| 635 |
|
The entire algorithm can be summarized as the sequential solution of |
| 636 |
|
the following equations: |
| 637 |
|
\begin{eqnarray} |
| 638 |
|
u^{*} & = & u^{n} + \Delta t G_u^{(n+1/2)} \label{eq:ustar-nh} \\ |
| 639 |
|
v^{*} & = & v^{n} + \Delta t G_v^{(n+1/2)} \label{eq:vstar-nh} \\ |
| 640 |
|
w^{*} & = & w^{n} + \Delta t G_w^{(n+1/2)} \label{eq:wstar-nh} \\ |
| 641 |
|
\eta^* & = & \epsilon_{fs} \left( \eta^{n} +P-E+R \right)- \Delta t |
| 642 |
|
\partial_x H \widehat{u^{*}} |
| 643 |
|
+ \partial_y H \widehat{v^{*}} |
| 644 |
|
\\ |
| 645 |
|
\partial_x g H \partial_x \eta^{n+1} |
| 646 |
|
+ \partial_y g H \partial_y \eta^{n+1} |
| 647 |
|
- \frac{\epsilon_{fs} \eta^{n+1}}{\Delta t^2} |
| 648 |
|
& = & |
| 649 |
|
- \frac{\eta^*}{\Delta t^2} |
| 650 |
|
\label{eq:elliptic-nh} |
| 651 |
|
\\ |
| 652 |
|
u^{**} & = & u^{*} - \Delta t g \partial_x \eta^{n+1} \label{eq:unx-nh}\\ |
| 653 |
|
v^{**} & = & v^{*} - \Delta t g \partial_y \eta^{n+1} \label{eq:vnx-nh}\\ |
| 654 |
|
\partial_{xx} \phi_{nh}^{n+1} + \partial_{yy} \phi_{nh}^{n+1} + |
| 655 |
|
\partial_{rr} \phi_{nh}^{n+1} & = & |
| 656 |
|
\partial_x u^{**} + \partial_y v^{**} + \partial_r w^{*} \\ |
| 657 |
|
u^{n+1} & = & u^{**} - \Delta t \partial_x \phi_{nh}^{n+1} \label{eq:un+1-nh}\\ |
| 658 |
|
v^{n+1} & = & v^{**} - \Delta t \partial_y \phi_{nh}^{n+1} \label{eq:vn+1-nh}\\ |
| 659 |
|
\partial_r w^{n+1} & = & - \partial_x u^{n+1} - \partial_y v^{n+1} |
| 660 |
|
\end{eqnarray} |
| 661 |
|
where the last equation is solved by vertically integrating for |
| 662 |
|
$w^{n+1}$. |
| 663 |
|
|
| 664 |
|
|
| 665 |
|
|
| 666 |
\section{Variants on the Free Surface} |
\section{Variants on the Free Surface} |
| 667 |
|
|
| 668 |
We now descibe the various formulations of the free-surface that |
We now describe the various formulations of the free-surface that |
| 669 |
include non-linear forms, implicit in time using Crank-Nicholson, |
include non-linear forms, implicit in time using Crank-Nicholson, |
| 670 |
explicit and [one day] split-explicit. First, we'll reiterate the |
explicit and [one day] split-explicit. First, we'll reiterate the |
| 671 |
underlying algorithm but this time using the notation consistent with |
underlying algorithm but this time using the notation consistent with |
| 672 |
the more general vertical coordinate $r$. The elliptic equation for |
the more general vertical coordinate $r$. The elliptic equation for |
| 673 |
free-surface coordinate (units of $r$), correpsonding to |
free-surface coordinate (units of $r$), corresponding to |
| 674 |
\ref{eq:discrete-time-backward-free-surface}, and |
\ref{eq:discrete-time-backward-free-surface}, and |
| 675 |
assuming no non-hydrostatic effects ($\epsilon_{nh} = 0$) is: |
assuming no non-hydrostatic effects ($\epsilon_{nh} = 0$) is: |
| 676 |
\begin{eqnarray} |
\begin{eqnarray} |
| 702 |
|
|
| 703 |
|
|
| 704 |
Once ${\eta}^{n+1}$ has been found, substituting into |
Once ${\eta}^{n+1}$ has been found, substituting into |
| 705 |
\ref{eq-tDsC-Hmom} yields $\vec{\bf v}^{n+1}$ if the model is |
\ref{eq:discrete-time-u,eq:discrete-time-v} yields $\vec{\bf v}^{n+1}$ if the model is |
| 706 |
hydrostatic ($\epsilon_{nh}=0$): |
hydrostatic ($\epsilon_{nh}=0$): |
| 707 |
$$ |
$$ |
| 708 |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
\vec{\bf v}^{n+1} = \vec{\bf v}^{*} |
| 712 |
This is known as the correction step. However, when the model is |
This is known as the correction step. However, when the model is |
| 713 |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
non-hydrostatic ($\epsilon_{nh}=1$) we need an additional step and an |
| 714 |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
additional equation for $\phi'_{nh}$. This is obtained by substituting |
| 715 |
\ref{eq-tDsC-Hmom} and \ref{eq-tDsC-Vmom} into |
\ref{eq:discrete-time-u-nh}, \ref{eq:discrete-time-v-nh} and \ref{eq:discrete-time-w-nh} |
| 716 |
\ref{eq-tDsC-cont}: |
into continuity: |
| 717 |
\begin{equation} |
\begin{equation} |
| 718 |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
\left[ {\bf \nabla}_h^2 + \partial_{rr} \right] {\phi'_{nh}}^{n+1} |
| 719 |
= \frac{1}{\Delta t} \left( |
= \frac{1}{\Delta t} \left( |
| 793 |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
{\it implicSurfPress}, {\it implicDiv2DFlow}. They are read from |
| 794 |
the main data file "{\it data}" and are set by default to 1,1. |
the main data file "{\it data}" and are set by default to 1,1. |
| 795 |
|
|
| 796 |
Equations \ref{eq-tDsC-Hmom} and \ref{eq-tDsC-eta} are modified as follows: |
Equations \ref{eq:ustar-backward-free-surface} -- |
| 797 |
|
\ref{eq:vn+1-backward-free-surface} are modified as follows: |
| 798 |
$$ |
$$ |
| 799 |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
\frac{ \vec{\bf v}^{n+1} }{ \Delta t } |
| 800 |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
+ {\bf \nabla}_h b_s [ \beta {\eta}^{n+1} + (1-\beta) {\eta}^{n} ] |
Parent Directory
| 
Revision Log
| 
Revision Graph
| 
Patch