Difference between revisions of "Horizontal Mixing"
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==Deviatory Stress Tensor== | ==Deviatory Stress Tensor== | ||
<wikitex>The horizontal components of the divergence of the stress tensor (Wajsowicz, 1993) in nondimesional, orthogonal curvilinear coordinates ($\xi$, $\eta$, $s$) with dimensional, spatially-varying metric factors ($\frac{1}{m}$, $\frac{1}{n}$, $H_{z}$) and velocity components ($u$, $v$, $\omega H_{z}$) are given by | <wikitex>The horizontal components of the divergence of the stress tensor ([[Bibliography#WajsowiczRC_1993a | Wajsowicz, 1993]]) in nondimesional, orthogonal curvilinear coordinates ($\xi$, $\eta$, $s$) with dimensional, spatially-varying metric factors ($\frac{1}{m}$, $\frac{1}{n}$, $H_{z}$) and velocity components ($u$, $v$, $\omega H_{z}$) are given by: | ||
$$ \eqalign { | $$ \eqalign { | ||
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{\sigma}_{\eta s} &= 2\,K_{M}\,e_{\eta s}, \cr} \eqno{(3)} $$ | {\sigma}_{\eta s} &= 2\,K_{M}\,e_{\eta s}, \cr} \eqno{(3)} $$ | ||
and the strain field is | and the strain field is: | ||
$$ \eqalign { | $$ \eqalign { | ||
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n\;H_{z} {\pder{\omega}{\eta}}. \cr} \eqno{(4)} $$ | n\;H_{z} {\pder{\omega}{\eta}}. \cr} \eqno{(4)} $$ | ||
Here, $A_{M}(\xi,\eta)$ and $K_{M}(\xi,\eta,s)$ are the spatially varying horizontal and vertical viscosity coefficients, respectively, and $\nu$ is another (very small, often neglected) horizontal viscosity coefficient. Notice that because of the generalized terrain-following vertical coordinates of ROMS, we need to transform the horizontal partial derivatives from constant | Here, $A_{M}(\xi,\eta)$ and $K_{M}(\xi,\eta,s)$ are the spatially varying horizontal and vertical viscosity coefficients, respectively, and $\nu$ is another (very small, often neglected) horizontal viscosity coefficient. Notice that because of the generalized terrain-following vertical coordinates of ROMS, we need to transform the horizontal partial derivatives from constant ''z-''surfaces to constant ''s-''surfaces. And the vertical metric or level thickness is the Jacobian of the transformation, $H_{z}={\pder{z}{s}}$. Also in these models, the ''vertical'' velocity is computed as $\frac{\omega H_{z}}{mn}$ and has units of $\hbox{m}^3/\hbox{s}$.</wikitex> | ||
===Transverse Stress Tensor=== | ===Transverse Stress Tensor=== | ||
<wikitex>Assuming transverse isotropy, as in Sadourny and Maynard (1997) and Griffies and Hallberg (2000), the deviatoric stress tensor can be split into vertical and horizontal sub-tensors. The horizontal (or transverse) sub-tensor is symmetric, it has a null trace, and it possesses axial symmetry in the local vertical direction. Then, transverse stress tensor can be derived from (1) and (2) yielding | <wikitex>Assuming transverse isotropy, as in [[Bibliography#SadournyR_1997a | Sadourny and Maynard (1997)]] and [[Bibliography#GriffiesSM_2000a | Griffies and Hallberg (2000)]], the deviatoric stress tensor can be split into vertical and horizontal sub-tensors. The horizontal (or transverse) sub-tensor is symmetric, it has a null trace, and it possesses axial symmetry in the local vertical direction. Then, transverse stress tensor can be derived from (1) and (2) yielding | ||
$$ \eqalign { | $$ \eqalign { | ||
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Notice the flux form of (5) and the symmetry between the $F^{u\xi}$ and $F^{v\eta}$ terms which are defined at density points on a C-grid. Similarly, the $F^{u\eta}$ and $F^{v\xi}$ terms are symmetric and defined at vorticity points. These staggering positions are optimal for the discretization of the tensor; it has no computational modes and satisfy first-moment conservation. | Notice the flux form of (5) and the symmetry between the $F^{u\xi}$ and $F^{v\eta}$ terms which are defined at density points on a C-grid. Similarly, the $F^{u\eta}$ and $F^{v\xi}$ terms are symmetric and defined at vorticity points. These staggering positions are optimal for the discretization of the tensor; it has no computational modes and satisfy first-moment conservation. | ||
The biharmonic friction operator can be computed by applying twice the tensor operator (5), but with the squared root of the biharmonic viscosity coefficient (Griffies and Hallberg, 2000). For simplicity and momentum balance, the thickness $H_{z}$ appears only when computing the second harmonic operator as in Griffies and Hallberg (2000).</wikitex> | The biharmonic friction operator can be computed by applying twice the tensor operator (5), but with the squared root of the biharmonic viscosity coefficient ([[Bibliography#GriffiesSM_2000a |Griffies and Hallberg, 2000]]). For simplicity and momentum balance, the thickness $H_{z}$ appears only when computing the second harmonic operator as in [[Bibliography#GriffiesSM_2000a | Griffies and Hallberg (2000)]].</wikitex> | ||
===Rotated Transverse Stress Tensor=== | ===Rotated Transverse Stress Tensor=== | ||
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Notice that transverse stress tensor remains invariant under coordinate transformation. The rotated tensor (7) retains the | Notice that transverse stress tensor remains invariant under coordinate transformation. The rotated tensor (7) retains the | ||
same properties as the unrotated tensor (5). The additional terms that arise from the slopes of $s$-surfaces along | same properties as the unrotated tensor (5). The additional terms that arise from the slopes of $s$-surfaces along | ||
geopotentials are discretized using a modified version of the triad approach of Griffies | geopotentials are discretized using a modified version of the triad approach of [[Bibliography#GriffiesSM_2000a | Griffies ''et al.'' (1998)]].</wikitex> | ||
==Radiation Stresses== | ==Radiation Stresses== | ||
Revision as of 20:21, 20 July 2008
Horizontal Mixing
Deviatory Stress Tensor
<wikitex>The horizontal components of the divergence of the stress tensor ( Wajsowicz, 1993) in nondimesional, orthogonal curvilinear coordinates ($\xi$, $\eta$, $s$) with dimensional, spatially-varying metric factors ($\frac{1}{m}$, $\frac{1}{n}$, $H_{z}$) and velocity components ($u$, $v$, $\omega H_{z}$) are given by:
$$ \eqalign {
F^{u} \equiv \widehat{\xi}\cdot\left(\nabla\cdot\vec{\sigma}\right) =
\frac{mn}{H_{z}} \Biggl[ & {\pder{}{\xi}} \Biggl( \frac{H_{z}{\sigma}_{\xi\xi}} {n} \Biggr) +
{\pder{}{\eta}} \Biggl( \frac{H_{z}{\sigma}_{\xi\eta}}{m} \Biggr) +
{\pder{}{s}} \Biggl( \frac{{\sigma}_{\xi s}}{mn} \Biggr) + \cr
&H_{z}{\sigma}_{\xi\eta} {\pder{}{\eta}} \left( \frac{1}{m}\right) -
H_{z}{\sigma}_{\eta\eta} {\pder{}{\xi}} \left( \frac{1}{n}\right) -
\frac{1}{n} {\sigma}_{ss}{\pder{H_{z}}{\xi}} \Biggr] \cr} \eqno{(1)} $$
$$ \eqalign {
F^{v} \equiv \widehat{\eta}\cdot\left(\nabla\cdot\vec{\sigma}\right) =
\frac{mn}{H_{z}} \Biggl[ & {\pder{}{\xi}} \Biggl( \frac{H_{z}{\sigma}_{\eta\xi}} {n} \Biggr) +
{\pder{}{\eta}} \Biggl( \frac{H_{z}{\sigma}_{\eta\eta}}{m} \Biggr) +
{\pder{}{s}} \Biggl( \frac{{\sigma}_{\eta s}}{mn} \Biggr) + \cr
&H_{z}{\sigma}_{\eta\xi} {\pder{}{\xi}} \left( \frac{1}{n} \right) -
H_{z}{\sigma}_{\xi\xi} {\pder{}{\eta}} \left( \frac{1}{m} \right) -
\frac{1}{m}{\sigma}_{ss} {\pder{H_{z}}{\eta}} \Biggr] \cr} \eqno{(2)} $$
where
$$ \eqalign {
{\sigma}_{\xi\xi} &= \left( A_{M} + \nu \right) e_{\xi\xi} + \left( \nu - A_{M}\right) e_{\eta\eta}, \cr
\noalign{\smallskip}
{\sigma}_{\eta\eta} &= \left( \nu - A_{M} \right) e_{\xi\xi} + \left( A_{M} + \nu\right) e_{\eta\eta}, \cr
\noalign{\smallskip}
{\sigma}_{ss} &= 2\,\nu\,e_{ss}, \cr
\noalign{\smallskip}
{\sigma}_{\xi\eta} &= {\sigma}_{\eta\xi} = 2\,A_{M}\,e_{\xi\eta}, \cr
\noalign{\smallskip}
{\sigma}_{\xi s} &= 2\,K_{M}\,e_{\xi s}, \cr
\noalign{\smallskip}
{\sigma}_{\eta s} &= 2\,K_{M}\,e_{\eta s}, \cr} \eqno{(3)} $$
and the strain field is:
$$ \eqalign {
e_{\xi\xi} &= m {\pder{u}{\xi}} + mnv {\pder{}{\eta}} \left( \frac{1}{m} \right), \cr
\noalign{\smallskip}
e_{\eta\eta} &= n\;{\pder{v}{\eta}} + mnu {\pder{}{\xi}} \left( \frac{1}{n} \right), \cr
\noalign{\smallskip}
e_{ss} &= \frac{1}{H_{z}} {\pder{\left( \omega H_{z} \right) }{s}} +
\frac{m}{H_{z}} u {\pder{H_{z}}{\xi}} +
\frac{n}{H_{z}} v {\pder{H_{z}}{\eta}}, \cr
\noalign{\smallskip}
2\,e_{\xi\eta} &= \frac{m}{n} {\pder{\left( nv \right) }{\xi}} +
\frac{n}{m} {\pder{\left( mu \right) }{\eta}}, \cr
\noalign{\smallskip}
2\,e_{\xi s} &= \frac{1}{mH_{z}} {\pder{\left( mu \right) }{s}} +
m H_{z} {\pder{\omega}{\xi}}, \cr
\noalign{\smallskip}
2\,e_{\eta s} &= \frac{1}{nH_{z}} \; {\pder{\left(nv\right)}{s}} \;+
n\;H_{z} {\pder{\omega}{\eta}}. \cr} \eqno{(4)} $$
Here, $A_{M}(\xi,\eta)$ and $K_{M}(\xi,\eta,s)$ are the spatially varying horizontal and vertical viscosity coefficients, respectively, and $\nu$ is another (very small, often neglected) horizontal viscosity coefficient. Notice that because of the generalized terrain-following vertical coordinates of ROMS, we need to transform the horizontal partial derivatives from constant z-surfaces to constant s-surfaces. And the vertical metric or level thickness is the Jacobian of the transformation, $H_{z}={\pder{z}{s}}$. Also in these models, the vertical velocity is computed as $\frac{\omega H_{z}}{mn}$ and has units of $\hbox{m}^3/\hbox{s}$.</wikitex>
Transverse Stress Tensor
<wikitex>Assuming transverse isotropy, as in Sadourny and Maynard (1997) and Griffies and Hallberg (2000), the deviatoric stress tensor can be split into vertical and horizontal sub-tensors. The horizontal (or transverse) sub-tensor is symmetric, it has a null trace, and it possesses axial symmetry in the local vertical direction. Then, transverse stress tensor can be derived from (1) and (2) yielding
$$ \eqalign {
H_{z}F^{u} &= {n^2}m {\partial\over\partial\xi}\left(\frac{H_{z}F^{u\xi}}{n}\right) +
{m^2}n {\partial\over\partial\eta}\left(\frac{H_{z}F^{u\eta}}{m}\right) \cr
\noalign{\smallskip}
H_{z}F^{v} &= {n^2}m {\partial\over\partial\xi}\left(\frac{H_{z}F^{v\xi}}{n}\right) +
{m^2}n {\partial\over\partial\eta}\left(\frac{H_{z}F^{v\eta}}{m}\right) \cr}
\eqno{(5)} $$
where
$$ \eqalign {
F^{u\xi} &= \frac{1}{n} \;A_{M}\left[
\frac{m}{n} {\pder{\left( nu \right) }{\xi}} \;-
\frac{n}{m} {\pder{\left( mv \right) }{\eta}} \right], \cr
\noalign{\smallskip}
F^{u\eta} &= \frac{1}{m} A_{M}\left[
\frac{n}{m} {\pder{\left( mu \right) }{\eta}} +
\frac{m}{n} {\pder{\left( nv \right) }{\xi}} \;\right], \cr
\noalign{\medskip}
F^{v\xi} &= \frac{1}{n} \;A_{M}\left[
\frac{m}{n} {\pder{\left( nv \right) }{\xi}} \;+
\frac{n}{m} {\pder{\left( mu \right) }{\eta}} \right], \cr
\noalign{\smallskip}
F^{v\eta} &= \frac{1}{m} A_{M}\left[
\frac{n}{m} {\pder{\left( mv \right) }{\eta}} -
\frac{m}{n} {\pder{\left( nu \right) }{\xi}} \;\right]. \cr}
\eqno{(6)} $$
Notice the flux form of (5) and the symmetry between the $F^{u\xi}$ and $F^{v\eta}$ terms which are defined at density points on a C-grid. Similarly, the $F^{u\eta}$ and $F^{v\xi}$ terms are symmetric and defined at vorticity points. These staggering positions are optimal for the discretization of the tensor; it has no computational modes and satisfy first-moment conservation.
The biharmonic friction operator can be computed by applying twice the tensor operator (5), but with the squared root of the biharmonic viscosity coefficient (Griffies and Hallberg, 2000). For simplicity and momentum balance, the thickness $H_{z}$ appears only when computing the second harmonic operator as in Griffies and Hallberg (2000).</wikitex>
Rotated Transverse Stress Tensor
<wikitex>In some applications with tall and steep topography, it will be advantageous to reduce substantially the contribution of the stress tensor (5) to the vertical mixing when operating along constant $s$-surfaces. The transverse stress tensor rotated along geopotentials (constant depth) is, then, given by
$$ \eqalign {
H_{z}R^{u} &= {n^2}m {\pder{}{\xi}} \Biggl( \frac{H_{z}R^{u\xi}} {n} \Biggr) +
{m^2}n {\pder{}{\eta}} \Biggl( \frac{H_{z}R^{u\eta}}{m} \Biggr) +
{\pder{}{s}} \Biggl( R^{us} \Biggr) \cr
\noalign{\smallskip}
H_{z}R^{v} &= {n^2}m {\pder{}{\xi}} \Biggl( \frac{H_{z}R^{v\xi}} {n} \Biggr) +
{m^2}n {\pder{}{\eta}} \Biggl( \frac{H_{z}R^{v\eta}}{m} \Biggr) +
{\pder{}{s}} \Biggl( R^{vs} \Biggr) \cr}
\eqno{(7)} $$
where
$$ \eqalign {
R^{u\xi} = &\frac{1}{n}\; A_{M} \left[
\frac{1}{n}\;\left( m {\pder{\left(nu\right)}{\xi}} -
m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nu \right) }{s}} \right) -
\frac{1}{m} \left( n {\pder{\left( mv \right) }{\eta}} -
n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mv \right) }{s}}\right)
\right], \cr
\noalign{\medskip}
R^{u\eta} = &\frac{1}{m} A_{M} \left[
\frac{1}{m} \left( n {\pder{\left( mu \right) }{\eta}} -
n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mu \right) }{s}} \right) +
\frac{1}{n}\;\left( m {\pder{\left( nv \right) }{\xi}} -
m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nv \right) }{s}} \right)
\right], \cr
\noalign{\medskip}
R^{us} = &m {\pder{z}{\xi}} A_{M} \left[
\frac{1}{n}\;\left( m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nu \right) }{s}} -
m {\pder{\left( nu \right) }{\xi}} \right) -
\frac{1}{m} \left( n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mv \right) }{s}} -
n {\pder{\left( mv \right) }{\eta}} \right)
\right] +\cr
&n\; {\pder{z}{\eta}} A_{M} \left[
\frac{1}{m} \left( n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mu \right) }{s}} -
n {\pder{\left( mu \right) }{\eta}} \right) +
\frac{1}{n}\;\left( m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nv \right) }{s}} -
m {\pder{\left( nv \right) }{\xi}} \right)
\right], \cr
\noalign{\bigskip}
R^{v\xi} = &\frac{1}{n}\;A_{M} \left[
\frac{1}{n}\;\left( m {\pder{\left( nv \right) }{\xi}} -
m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nv \right) }{s}} \right) +
\frac{1}{m} \left( n {\pder{\left( mu \right) }{\eta}}-
n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mu \right) }{s}} \right)
\right], \cr
\noalign{\medskip}
R^{v\eta} = &\frac{1}{m}A_{M} \left[
\frac{1}{m} \left( n {\pder{\left( mv \right) }{\eta}} -
n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mv \right) }{s}} \right) -
\frac{1}{n}\;\left( m {\pder{\left( nu \right) }{\xi}} -
m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nu \right )}{s}} \right)
\right], \cr
\noalign{\medskip}
R^{vs} = &m {\pder{z}{\xi}} A_{M} \left[
\frac{1}{n}\;\left( m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nv \right) }{s}} -
m {\pder{\left( nv \right) }{\xi}} \right) +
\frac{1}{m} \left( n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mu \right) }{s}} -
n {\pder{\left( mu \right) }{\eta}} \right)
\right] +\cr
&n\; {\pder{z}{\eta}} A_{M} \left[
\frac{1}{m} \left( n {\pder{z}{\eta}} \frac{1}{H_{z}}
{\pder{\left( mv \right) }{s}} -
n {\pder{\left( mv \right) }{\eta}} \right) -
\frac{1}{n}\;\left( m {\pder{z}{\xi}} \frac{1}{H_{z}}
{\pder{\left( nu \right) }{s}} -
m {\pder{\left( nu \right) }{\xi}} \right)
\right]. \cr} \eqno{(8)} $$
Notice that transverse stress tensor remains invariant under coordinate transformation. The rotated tensor (7) retains the same properties as the unrotated tensor (5). The additional terms that arise from the slopes of $s$-surfaces along geopotentials are discretized using a modified version of the triad approach of Griffies et al. (1998).</wikitex>
