Equations of Motion - Revision history

Robertson: /* Vertical Boundary Conditions */

2024-10-21T15:03:14Z

Vertical Boundary Conditions

← Older revision		Revision as of 15:03, 21 October 2024
Line 135:		Line 135:
	{\| class="eqno"		{\| class="eqno"
	\|<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t) \\		\|<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t) \\
	&K_m \, \frac{\partial v}{\partial z} = \tau_b^y (x,y,t) \\		&K_m \, \frac{\partial v}{\partial z} = \tau_b^y(x,y,t) \\
	&K_C \, \frac{\partial C}{\partial z} = 0 \\		&K_C \, \frac{\partial C}{\partial z} = 0 \\
	&- w + \vec{v} \cdot \nabla h = 0 \end{align}</math> <!-- \eqno{(9)}-->		&-w + \vec{v} \cdot \nabla h = 0 \end{align}</math> <!-- \eqno{(9)}-->
	\|(9)		\|(9)
	\|}		\|}

Wilkin at 17:05, 20 January 2022

2022-01-20T17:05:16Z

← Older revision		Revision as of 17:05, 20 January 2022
Line 103:		Line 103:
	\|}		\|}

	In the ocean, vertical mixing due to molecular viscosity is extremely weak compared to the turbulent ~~Reynolds stresses~~, so the terms involving <math>\nu</math> and <math>\nu_\theta</math> can be neglected.		In the ocean, vertical mixing due to molecular viscosity is extremely weak compared to the turbulent mixing, so the terms involving <math>\nu</math> and <math>\nu_\theta</math> can be neglected.
	~~These~~ equations ~~are closed by parameterizing~~ the Reynolds stresses and turbulent tracer fluxes as:
			To close these equations, parameterizations of the Reynolds stresses and turbulent tracer fluxes are introduced as functions of the other variables and vertical turbulent eddy viscosity and eddy diffusivity coefficients, <math>K_M</math> and <math>K_C</math>, respectively.

	{\| class="eqno"		{\| class="eqno"
Line 113:		Line 114:
	\|}		\|}

	An overbar represents a time average and a prime represents a		An overbar represents a time average and a prime represents a fluctuation about the mean.
	fluctuation about the mean.
			Details on ROMS options for computing <math>K_M</math> and <math>K_C</math> are described in [[Vertical Mixing Parameterizations]].

	==Vertical Boundary Conditions==		==Vertical Boundary Conditions==

Wilkin at 16:59, 20 January 2022

2022-01-20T16:59:13Z

← Older revision		Revision as of 16:59, 20 January 2022
Line 52:		Line 52:
	{\| border="1" cellspacing="0" cellpadding="5" align="center"		{\| border="1" cellspacing="0" cellpadding="5" align="center"
	\| <math>{\cal D}_u, {\cal D}_v, {\cal D}_C</math>		\| <math>{\cal D}_u, {\cal D}_v, {\cal D}_C</math>
	\|style="width: 350px;"\| diffusive terms		\|style="width: 350px;"\| horizontal diffusive terms
	\|-		\|-
	\| <math>{\cal F}_u, {\cal F}_v, {\cal F}_C</math>		\| <math>{\cal F}_u, {\cal F}_v, {\cal F}_C</math>
Line 103:		Line 103:
	\|}		\|}

			In the ocean, vertical mixing due to molecular viscosity is extremely weak compared to the turbulent Reynolds stresses, so the terms involving <math>\nu</math> and <math>\nu_\theta</math> can be neglected.
	These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:		These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:

Robertson at 13:35, 4 August 2015

2015-08-04T13:35:54Z

← Older revision		Revision as of 13:35, 4 August 2015
Line 3:		Line 3:
	The primitive equations in Cartesian coordinates are shown here. The momentum balance in the <math>x</math>- and <math>y</math>-directions are:		The primitive equations in Cartesian coordinates are shown here. The momentum balance in the <math>x</math>- and <math>y</math>-directions are:

	<math display="block">\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u - fv = - \frac{\partial \phi}{\partial x} - {\partial \over \partial z} \left( \overline{u'w'} - \nu {\partial u \over \partial z} \right)+ {\cal F}_u + {\cal D}_u </math>		{\| class="eqno"
			\|<math display="block">\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u - fv = - \frac{\partial \phi}{\partial x} - {\partial \over \partial z} \left( \overline{u'w'} - \nu {\partial u \over \partial z} \right)+ {\cal F}_u + {\cal D}_u </math><!--\eqno{(1)}-->
			\|(1)
			\|}

	<math display="block">\frac{\partial v}{\partial t} + \vec{v} \cdot \nabla v + fu = - \frac{\partial \phi}{\partial y} - {\partial \over \partial z} \left( \overline{v'w'} - \nu {\partial v \over \partial z} \right) + {\cal F}_v + {\cal D}_v </math>		{\| class="eqno"
			\|<math display="block">\frac{\partial v}{\partial t} + \vec{v} \cdot \nabla v + fu = - \frac{\partial \phi}{\partial y} - {\partial \over \partial z} \left( \overline{v'w'} - \nu {\partial v \over \partial z} \right) + {\cal F}_v + {\cal D}_v </math><!--\eqno{(2)}-->
			\|(2)
			\|}

	The time evolution of a scalar concentration field, <math>C(x,y,z,t)</math> (e.g. salinity, temperature, or nutrients), is governed by the advective-diffusive equation:		The time evolution of a scalar concentration field, <math>C(x,y,z,t)</math> (e.g. salinity, temperature, or nutrients), is governed by the advective-diffusive equation:

	<math display="block">\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = -{\partial \over \partial z} \left( \overline{C'w'} - \nu_\theta {\partial C \over \partial z} \right) + {\cal F}_C + {\cal D}_C </math> <!--\eqno{(3)}-->		{\| class="eqno"
			\|<math display="block">\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = -{\partial \over \partial z} \left( \overline{C'w'} - \nu_\theta {\partial C \over \partial z} \right) + {\cal F}_C + {\cal D}_C </math> <!--\eqno{(3)}-->
			\|(3)
			\|}
	The equation of state is given by:		The equation of state is given by:

	<math display="block">\rho = \rho(T,S,P) </math><!--\eqno{(4)}-->		{\| class="eqno"
			\|<math display="block">\rho = \rho(T,S,P) </math><!--\eqno{(4)}-->
			\|(4)
			\|}

	In the Boussinesq approximation, density variations are neglected in the momentum equations except in their		In the Boussinesq approximation, density variations are neglected in the momentum equations except in their
Line 21:		Line 32:
	force:		force:

	<math display="block">\frac{\partial \phi}{\partial z} = -\frac{\rho g}{\rho_o} </math><!--\eqno{(5)}-->		{\| class="eqno"
			\|<math display="block">\frac{\partial \phi}{\partial z} = -\frac{\rho g}{\rho_o} </math><!--\eqno{(5)}-->
			\|(5)
			\|}

	The final equation expresses the continuity equation		The final equation expresses the continuity equation
	for an incompressible fluid:		for an incompressible fluid:

	<math display="block">\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 </math><!--\eqno{(6)}-->		{\| class="eqno"
			\|<math display="block">\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 </math><!--\eqno{(6)}-->
			\|(6)
			\|}

	For the moment, the effects of forcing		For the moment, the effects of forcing
Line 84:		Line 101:
	\| <math>\zeta(x,y,t)</math>		\| <math>\zeta(x,y,t)</math>
	\| the surface elevation		\| the surface elevation
	\|-
	\|}		\|}

	These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:		These equations are closed by parameterizing the Reynolds stresses and turbulent tracer fluxes as:
	<math display="block">\overline{u'w'} = -K_M {\partial u \over \partial z};\qquad
			{\| class="eqno"
			\|<math display="block">\overline{u'w'} = -K_M {\partial u \over \partial z};\qquad
	\overline{v'w'} = -K_M {\partial v \over \partial z};\qquad		\overline{v'w'} = -K_M {\partial v \over \partial z};\qquad
	\overline{C'w'} = -K_C {\partial C \over \partial z}. </math> <!--\eqno{(7)}-->		\overline{C'w'} = -K_C {\partial C \over \partial z}. </math> <!--\eqno{(7)}-->
			\|(7)
			\|}

	An overbar represents a time average and a prime represents a		An overbar represents a time average and a prime represents a
	fluctuation about the mean.		fluctuation about the mean.
Line 98:		Line 119:

	:top (<math>z = \zeta(x,y,t))</math>:		:top (<math>z = \zeta(x,y,t))</math>:
	<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau^x_s (x,y,t) \\
			{\| class="eqno"
			\|<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau^x_s (x,y,t) \\
	&K_m \, \frac{\partial v}{\partial z} = \tau_s^y(x,y,t) \\		&K_m \, \frac{\partial v}{\partial z} = \tau_s^y(x,y,t) \\
	&K_C \, \frac{\partial C}{\partial z} = {Q_C \over \rho_o c_P} \\		&K_C \, \frac{\partial C}{\partial z} = {Q_C \over \rho_o c_P} \\
	&w = {\partial \zeta \over \partial t} \end{align}</math> <!--\eqno{(8)}-->		&w = {\partial \zeta \over \partial t} \end{align}</math> <!--\eqno{(8)}-->
			\|(8)
			\|}

	:and bottom (<math>z = -h(x,y)</math>):		:and bottom (<math>z = -h(x,y)</math>):
	<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t) \\
			{\| class="eqno"
			\|<math display="block">\begin{align}&K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t) \\
	&K_m \, \frac{\partial v}{\partial z} = \tau_b^y (x,y,t) \\		&K_m \, \frac{\partial v}{\partial z} = \tau_b^y (x,y,t) \\
	&K_C \, \frac{\partial C}{\partial z} = 0 \\		&K_C \, \frac{\partial C}{\partial z} = 0 \\
	&- w + \vec{v} \cdot \nabla h = 0 \end{align}</math> <!-- \eqno{(9)}-->		&- w + \vec{v} \cdot \nabla h = 0 \end{align}</math> <!-- \eqno{(9)}-->
			\|(9)
			\|}

	The new variables above are:		The new variables above are:

Robertson at 05:27, 31 July 2015

2015-07-31T05:27:07Z

← Older revision		Revision as of 05:27, 31 July 2015
Line 35:		Line 35:
	{\| border="1" cellspacing="0" cellpadding="5" align="center"		{\| border="1" cellspacing="0" cellpadding="5" align="center"
	\| <math>{\cal D}_u, {\cal D}_v, {\cal D}_C</math>		\| <math>{\cal D}_u, {\cal D}_v, {\cal D}_C</math>
	\| diffusive terms		\|style="width: 350px;"\| diffusive terms
	\|-		\|-
	\| <math>{\cal F}_u, {\cal F}_v, {\cal F}_C</math>		\| <math>{\cal F}_u, {\cal F}_v, {\cal F}_C</math>

Robertson at 05:21, 31 July 2015

2015-07-31T05:21:36Z

Show changes

Kate: equation numbers

2008-07-24T21:15:27Z

equation numbers

← Older revision		Revision as of 21:15, 24 July 2008
Line 3:		Line 3:
	<wikitex>The primitive equations in Cartesian coordinates are shown here. The momentum balance in the $x$- and $y$-directions are:		<wikitex>The primitive equations in Cartesian coordinates are shown here. The momentum balance in the $x$- and $y$-directions are:

	$$\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u - fv = - \frac{\partial \phi}{\partial x} - {\partial \over \partial z} \left( \overline{u'w'} - \nu {\partial u \over \partial z} \right)+ {\cal F}_u + {\cal D}_u$$		$$\frac{\partial u}{\partial t} + \vec{v} \cdot \nabla u - fv = - \frac{\partial \phi}{\partial x} - {\partial \over \partial z} \left( \overline{u'w'} - \nu {\partial u \over \partial z} \right)+ {\cal F}_u + {\cal D}_u \eqno{(1)}$$

	$$\frac{\partial v}{\partial t} + \vec{v} \cdot \nabla v + fu = - \frac{\partial \phi}{\partial y} - {\partial \over \partial z} \left( \overline{v'w'} - \nu {\partial v \over \partial z} \right) + {\cal F}_v + {\cal D}_v$$		$$\frac{\partial v}{\partial t} + \vec{v} \cdot \nabla v + fu = - \frac{\partial \phi}{\partial y} - {\partial \over \partial z} \left( \overline{v'w'} - \nu {\partial v \over \partial z} \right) + {\cal F}_v + {\cal D}_v \eqno{(2)}$$

	The time evolution of a scalar concentration field, $C(x,y,z,t)$ (e.g. salinity, temperature, or nutrients), is governed by the advective-diffusive equation:		The time evolution of a scalar concentration field, $C(x,y,z,t)$ (e.g. salinity, temperature, or nutrients), is governed by the advective-diffusive equation:

	$$\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = -{\partial \over \partial z} \left( \overline{C'w'} - \nu_\theta {\partial C \over \partial z} \right) + {\cal F}_C + {\cal D}_C$$		$$\frac{\partial C}{\partial t} + \vec{v} \cdot \nabla C = -{\partial \over \partial z} \left( \overline{C'w'} - \nu_\theta {\partial C \over \partial z} \right) + {\cal F}_C + {\cal D}_C \eqno{(3)}$$

	The equation of state is given by:		The equation of state is given by:

	$$\rho = \rho(T,S,P)$$		$$\rho = \rho(T,S,P) \eqno{(4)}$$

	In the Boussinesq approximation, density variations are neglected in the momentum equations except in their		In the Boussinesq approximation, density variations are neglected in the momentum equations except in their
Line 21:		Line 21:
	force:		force:

	$$\frac{\partial \phi}{\partial z} = -\frac{\rho g}{\rho_o}$$		$$\frac{\partial \phi}{\partial z} = -\frac{\rho g}{\rho_o} \eqno{(5)}$$

	The final equation expresses the continuity equation		The final equation expresses the continuity equation
	for an incompressible fluid:		for an incompressible fluid:

	$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$		$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \eqno{(6)}$$

	For the moment, the effects of forcing		For the moment, the effects of forcing
Line 90:		Line 90:
	\overline{u'w'} = -K_M {\partial u \over \partial z};\qquad		\overline{u'w'} = -K_M {\partial u \over \partial z};\qquad
	\overline{v'w'} = -K_M {\partial v \over \partial z};\qquad		\overline{v'w'} = -K_M {\partial v \over \partial z};\qquad
	\overline{C'w'} = -K_C {\partial C \over \partial z}.$$		\overline{C'w'} = -K_C {\partial C \over \partial z}. \eqno{(7)}$$
	An overbar represents a time average and a prime represents a		An overbar represents a time average and a prime represents a
	fluctuation about the mean.		fluctuation about the mean.
Line 99:		Line 99:

	:top ($z = \zeta(x,y,t))$:		:top ($z = \zeta(x,y,t))$:
	::$K_m \, \frac{\partial u}{\partial z} = \tau^x_s (x,y,t)$		$$ \eqalign{ &K_m \, \frac{\partial u}{\partial z} = \tau^x_s (x,y,t) \cr
	~~::$~~K_m \, \frac{\partial v}{\partial z} = \tau_s^y(x,y,t)$		&K_m \, \frac{\partial v}{\partial z} = \tau_s^y(x,y,t) \cr
	~~::$~~K_C \, \frac{\partial C}{\partial z} = {Q_C \over \rho_o c_P}$		&K_C \, \frac{\partial C}{\partial z} = {Q_C \over \rho_o c_P} \cr
	~~::$~~w = {\partial \zeta \over \partial t}$		&w = {\partial \zeta \over \partial t} \cr} \eqno{(8)}$$

	:and bottom ($z = -h(x,y)$):		:and bottom ($z = -h(x,y)$):
	::$K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t)$		$$ \eqalign{ &K_m \, \frac{\partial u}{\partial z} = \tau_b^x (x,y,t) \cr
	~~::$~~K_m \, \frac{\partial v}{\partial z} = \tau_b^y (x,y,t)$		&K_m \, \frac{\partial v}{\partial z} = \tau_b^y (x,y,t) \cr
	~~::$~~K_C \, \frac{\partial C}{\partial z} = 0$		&K_C \, \frac{\partial C}{\partial z} = 0 \cr
	~~::$~~- w + \vec{v} \cdot \nabla h = 0$		&- w + \vec{v} \cdot \nabla h = 0 \cr} \eqno{(9)}$$

	The new variables above are:		The new variables above are:

Kate at 00:30, 22 July 2008

2008-07-22T00:30:45Z

← Older revision		Revision as of 00:30, 22 July 2008
Line 34:		Line 34:
	The variables used are shown here:		The variables used are shown here:
	{\| border="1" cellspacing="0" cellpadding="5" align="center"		{\| border="1" cellspacing="0" cellpadding="5" align="center"
	\| ${\cal D}_u, {\cal D}_v, {\cal D}~~_T, {\cal D}_S~~$		\| ${\cal D}_u, {\cal D}_v, {\cal D}_C$
	\| diffusive terms		\| diffusive terms
	\|-		\|-
	\| ${\cal F}_u, {\cal F}_v, {\cal F}~~_T, {\cal F}_S~~$		\| ${\cal F}_u, {\cal F}_v, {\cal F}_C$
	\| forcing terms		\| forcing terms
	\|-		\|-

Kate: added vertical mixing term from Haidvogel et al.

2008-07-22T00:29:44Z

added vertical mixing term from Haidvogel et al.

Show changes

Arango at 18:26, 11 July 2008

2008-07-11T18:26:24Z

← Older revision		Revision as of 18:26, 11 July 2008
Line 1:		Line 1:
	<div class="title">Equations of Motion</div>		<div class="title">Equations of Motion</div>
	<wikitex>		<wikitex>

	The primitive equations in Cartesian coordinates are shown here. The momentum balance in the $x$- and $y$-directions are:		The primitive equations in Cartesian coordinates are shown here. The momentum balance in the $x$- and $y$-directions are: