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Changeset 11335 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex – NEMO

# Changeset 11335 for NEMO/trunk/doc/latex/NEMO/subfiles/annex_A.tex

Ignore:
Timestamp:
2019-07-24T12:16:18+02:00 (4 years ago)
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review of chap_model_basics, annex_A and annex_B

File:
1 edited

### Legend:

Unmodified
 r11151 \label{apdx:A_s_slope} \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s \sigma_1 =\frac{1}{e_1 } \; \left. {\frac{\partial z}{\partial i}} \right|_s \quad \text{and} \quad \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s The chain rule to establish the model equations in the curvilinear $s-$coordinate system is: \sigma_2 =\frac{1}{e_2 } \; \left. {\frac{\partial z}{\partial j}} \right|_s . The model fields (e.g. pressure $p$) can be viewed as functions of $(i,j,z,t)$ (e.g. $p(i,j,z,t)$) or as functions of $(i,j,s,t)$ (e.g. $p(i,j,s,t)$). The symbol $\bullet$ will be used to represent any one of these fields.  Any infinitesimal'' change in $\bullet$ can be written in two forms: \label{apdx:A_s_infin_changes} \begin{aligned} & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,s,t} + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,s,t} + \delta s \left. \frac{ \partial \bullet }{\partial s} \right|_{i,j,t} + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,s} , \\ & \delta \bullet =  \delta i \left. \frac{ \partial \bullet }{\partial i} \right|_{j,z,t} + \delta j \left. \frac{ \partial \bullet }{\partial i} \right|_{i,z,t} + \delta z \left. \frac{ \partial \bullet }{\partial z} \right|_{i,j,t} + \delta t \left. \frac{ \partial \bullet }{\partial t} \right|_{i,j,z} . \end{aligned} Using the first form and considering a change $\delta i$ with $j, z$ and $t$ held constant, shows that \label{apdx:A_s_chain_rule} \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,z,t}  = \left. {\frac{\partial \bullet }{\partial i}} \right|_{j,s,t} + \left. {\frac{\partial s       }{\partial i}} \right|_{j,z,t} \; \left. {\frac{\partial \bullet }{\partial s}} \right|_{i,j,t} . The term $\left. \partial s / \partial i \right|_{j,z,t}$ can be related to the slope of constant $s$ surfaces, (\autoref{apdx:A_s_slope}), by applying the second of (\autoref{apdx:A_s_infin_changes}) with $\bullet$ set to $s$ and $j, t$ held constant \label{apdx:a_delta_s} \delta s|_{j,t} = \delta i \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} + \delta z \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} . Choosing to look at a direction in the $(i,z)$ plane in which $\delta s = 0$ and using (\autoref{apdx:A_s_slope}) we obtain \left. \frac{ \partial s }{\partial i} \right|_{j,z,t} = -  \left. \frac{ \partial z }{\partial i} \right|_{j,s,t} \; \left. \frac{ \partial s }{\partial z} \right|_{i,j,t} = - \frac{e_1 }{e_3 }\sigma_1  . \label{apdx:a_ds_di_z} Another identity, similar in form to (\autoref{apdx:a_ds_di_z}), can be derived by choosing $\bullet$ to be $s$ and using the second form of (\autoref{apdx:A_s_infin_changes}) to consider changes in which $i , j$ and $s$ are constant. This shows that \label{apdx:A_w_in_s} w_s = \left. \frac{ \partial z }{\partial t} \right|_{i,j,s} = - \left. \frac{ \partial z }{\partial s} \right|_{i,j,t} \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} = - e_3 \left. \frac{ \partial s }{\partial t} \right|_{i,j,z} . In what follows, for brevity, indication of the constancy of the $i, j$ and $t$ indices is usually omitted. Using the arguments outlined above one can show that the chain rules needed to establish the model equations in the curvilinear $s-$coordinate system are: \label{apdx:A_s_chain_rule} \begin{aligned} &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  = \left. {\frac{\partial \bullet }{\partial t}} \right|_s -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\ \left. {\frac{\partial \bullet }{\partial t}} \right|_s + \frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial t} , \\ &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  = \left. {\frac{\partial \bullet }{\partial i}} \right|_s -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}= \left. {\frac{\partial \bullet }{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\ +\frac{\partial \bullet }{\partial s}\; \frac{\partial s}{\partial i}= \left. {\frac{\partial \bullet }{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} , \\ &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  = \left. {\frac{\partial \bullet }{\partial j}} \right|_s - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= \left. {\frac{\partial \bullet }{\partial j}} \right|_s - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\ &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \left. {\frac{\partial \bullet }{\partial j}} \right|_s + \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}= \left. {\frac{\partial \bullet }{\partial j}} \right|_s - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} , \\ &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} . \end{aligned} In particular applying the time derivative chain rule to $z$ provides the expression for $w_s$, the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate: \label{apdx:A_w_in_s} w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} = e_3 \, \frac{\partial s}{\partial t} % ================================================================ \end{subequations} Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega$, defined as the volume flux across the moving $s$-surfaces per unit horizontal area: Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Using the first form of (\autoref{apdx:A_s_infin_changes}) and the definitions (\autoref{apdx:A_s_slope}) and (\autoref{apdx:A_w_in_s}) for $\sigma_1$, $\sigma_2$ and  $w_s$, one can show that the vertical velocity, $w_p$ of a point moving with the horizontal velocity of the fluid along an $s$ surface is given by \label{apdx:A_w_p} \begin{split} w_p  = & \left. \frac{ \partial z }{\partial t} \right|_s + \frac{u}{e_1} \left. \frac{ \partial z }{\partial i} \right|_s + \frac{v}{e_2} \left. \frac{ \partial z }{\partial j} \right|_s \\ = & w_s + u \sigma_1 + v \sigma_2 . \end{split} The vertical velocity across this surface is denoted by \label{apdx:A_w_s} \omega  = w - w_s - \sigma_1 \,u - \sigma_2 \,v    \\ with $w_s$ given by \autoref{apdx:A_w_in_s}, we obtain the expression for the divergence of the velocity in the curvilinear $s-$coordinate system: \begin{subequations} \begin{align*} { \begin{array}{*{20}l} \nabla \cdot {\mathrm {\mathbf U}} &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ \omega  = w - w_p = w - ( w_s + \sigma_1 \,u + \sigma_2 \,v )  . Hence \frac{1}{e_3 } \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right] = \frac{1}{e_3 } \frac{\partial}{\partial s} \left[  \omega + w_s \right] = \frac{1}{e_3 } \left[ \frac{\partial \omega}{\partial s} + \left. \frac{ \partial }{\partial t} \right|_s \frac{\partial z}{\partial s} \right] = \frac{1}{e_3 } \frac{\partial \omega}{\partial s} + \frac{1}{e_3 } \left. \frac{ \partial e_3}{\partial t} . \right|_s Using (\autoref{apdx:A_w_s}) in our expression for $\nabla \cdot {\mathrm {\mathbf U}}$ we obtain our final expression for the divergence of the velocity in the curvilinear $s-$coordinate system: \nabla \cdot {\mathrm {\mathbf U}} = \frac{1}{e_1 \,e_2 \,e_3 }    \left[ \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} + \frac{1}{e_3 } \frac{\partial w_s       }{\partial s} \\ \\ &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} + \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right) \\ \\ &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} + \frac{\partial}{\partial s} \frac{\partial s}{\partial t} + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\ &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right] + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} + \frac{1}{e_3 } \frac{\partial e_3}{\partial t} \end{array} } \end{align*} \end{subequations} + \frac{1}{e_3 } \left. \frac{\partial e_3}{\partial t} \right|_s . As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$coordinates is: {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right] +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 A additional term has appeared that take into account +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0 . An additional term has appeared that takes into account the contribution of the time variation of the vertical coordinate to the volume budget. + w \;\frac{\partial u}{\partial z} \\ \\ &= \left. {\frac{\partial u }{\partial t}} \right|_z - \left. \zeta \right|_z v +  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z -  \frac{1}{e_1 \,e_2 }\left[ { \left.{ \frac{\partial (e_2 \,v)}{\partial i} }\right|_z -\left.{ \frac{\partial (e_1 \,u)}{\partial j} }\right|_z } \right] \; v +  \frac{1}{2e_1} \left.{ \frac{\partial (u^2+v^2)}{\partial i} } \right|_z } \\ \\ &= \left. {\frac{\partial u }{\partial t}} \right|_z + \left. \zeta \right|_s \;v - \left. \zeta \right|_s \;v + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ &\qquad \qquad \qquad \quad + \frac{w}{e_3 } \;\frac{\partial u}{\partial s} - \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} + \left[   {\frac{\sigma_1 }{e_3 }\frac{\partial v}{\partial s} - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v - \frac{\sigma_1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} \\ \\ &= \left. {\frac{\partial u }{\partial t}} \right|_z + \left. \zeta \right|_s \;v - \left. \zeta \right|_s \;v + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ &\qquad \qquad \qquad \quad - \sigma_1 u\frac{\partial u}{\partial s} - \sigma_1 v\frac{\partial v}{\partial s}} \right] \\ \\ &= \left. {\frac{\partial u }{\partial t}} \right|_z + \left. \zeta \right|_s \;v - \left. \zeta \right|_s \;v + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s + \frac{1}{e_3} \left[  w - \sigma_2 v - \sigma_1 u  \right] \; \frac{\partial u}{\partial s}   \\ \; \frac{\partial u}{\partial s} .  \\ % \intertext{Introducing $\omega$, the dia-a-surface velocity given by (\autoref{apdx:A_w_s}) } \intertext{Introducing $\omega$, the dia-s-surface velocity given by (\autoref{apdx:A_w_s}) } % &= \left. {\frac{\partial u }{\partial t}} \right|_z + \left. \zeta \right|_s \;v - \left. \zeta \right|_s \;v + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\ + \frac{1}{e_3 } \left( \omega + w_s \right) \frac{\partial u}{\partial s}   \\ \end{array} } { \begin{array}{*{20}l} w_s  \;\frac{\partial u}{\partial s} = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s} = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , \frac{w_s}{e_3}  \;\frac{\partial u}{\partial s} = - \left. \frac{\partial s}{\partial t} \right|_z \;  \frac{\partial u }{\partial s} = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \ . \end{array} } \] leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, This leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, \ie the total $s-$coordinate time derivative : \begin{align} \left. \frac{D u}{D t} \right|_s = \left. {\frac{\partial u }{\partial t}} \right|_s + \left. \zeta \right|_s \;v - \left. \zeta \right|_s \;v + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s} . \end{align} Therefore, the vector invariant form of the total time derivative has exactly the same mathematical form in + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ &&- \frac{v}{e_1 e_2 }\left(    v \;\frac{\partial e_2 }{\partial i} -u  \;\frac{\partial e_1 }{\partial j}  \right) \\ -u  \;\frac{\partial e_1 }{\partial j}  \right) . \\ \end{array} } -  e_2 u \;\frac{\partial e_3 }{\partial i} -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right) -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] \\ \\ && - \frac{v}{e_1 e_2 }\left(   v  \;\frac{\partial e_2 }{\partial i} -u  \;\frac{\partial e_1 }{\partial j}   \right) \\ \\ && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} + \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right) -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right] - \frac{v}{e_1 e_2 }\left(   v   \;\frac{\partial e_2 }{\partial i} -u   \;\frac{\partial e_1 }{\partial j}  \right)                  \\ -u   \;\frac{\partial e_1 }{\partial j}  \right)     .             \\ % \intertext {Introducing a more compact form for the divergence of the momentum fluxes, + \left.  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right)    \right|_s - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} -u  \;\frac{\partial e_1 }{\partial j}            \right) -u  \;\frac{\partial e_1 }{\partial j}            \right). \end{flalign} which is the total time derivative expressed in the curvilinear $s-$coordinate system. & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\ & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\ &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1 . \end{split} \] Applying similar manipulation to the second component and replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes: replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it becomes: \label{apdx:A_grad_p_1} -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z &=-\frac{1}{\rho_o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\ + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) . \\ \end{split} $\begin{split} p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left( \rho_o \, d + 1 \right) \; e_3 \; dk \\ &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \rho_o \left( d + 1 \right) \; e_3 \; dk \\ &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + \rho_o g \, \int_z^\eta e_3 \; dk . \end{split}$ \label{apdx:A_pressure} p = \rho_o \; p_h' + g \, ( z + \eta ) p = \rho_o \; p_h' + \rho_o \, g \, ( \eta - z ) and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is: $\frac{\partial p_h'}{\partial k} = - d \, g \, e_3 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 .$ Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and using the definition of the density anomaly it comes the expression in two parts: using the definition of the density anomaly it becomes an expression in two parts: \label{apdx:A_grad_p_2} -\frac{1}{\rho_o \, e_1 } \left. {\frac{\partial p}{\partial i}} \right|_z &=-\frac{1}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\ + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} ,  \\ % -\frac{1}{\rho_o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z &=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\ + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j} . \\ \end{split} -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} +   D_u^{\vect{U}}  +   F_u^{\vect{U}} +   D_u^{\vect{U}}  +   F_u^{\vect{U}} , \end{multline} \begin{multline} -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} +  D_v^{\vect{U}}  +   F_v^{\vect{U}} +  D_v^{\vect{U}}  +   F_v^{\vect{U}} . \end{multline} \end{subequations} \label{apdx:A_PE_dyn_flux_u} \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} = \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) - \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,u}   \right) +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,v     \\ -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right) -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i} +   D_u^{\vect{U}}  +   F_u^{\vect{U}} +   D_u^{\vect{U}}  +   F_u^{\vect{U}} , \end{multline} \begin{multline} \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}= -  \nabla \cdot \left(   {{\mathrm {\mathbf U}}\,v}   \right) +   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} -   \left\{ {f + \frac{1}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i} -u  \;\frac{\partial e_1 }{\partial j}            \right)} \right\} \,u     \\ -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right) -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j} +  D_v^{\vect{U}}  +   F_v^{\vect{U}} +  D_v^{\vect{U}}  +   F_v^{\vect{U}} . \end{multline} \end{subequations} \label{apdx:A_dyn_zph} \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 \frac{\partial p_h'}{\partial k} = - d \, g \, e_3 . \left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) +   \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)               \right]       \\ +  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right) -  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right) +  D^{T} +F^{T} \end{multline} The expression for the advection term is a straight consequence of (A.4), The expression for the advection term is a straight consequence of (\autoref{apdx:A_sco_Continuity}), the expression of the 3D divergence in the $s-$coordinates established above.