Hydrostatic balance
During the setup-phase of a run, CGCAM will compute the thermodynamic
state such that the hydrostatic balance is satisfied numerically on
a particular point distribution in \(z\). This point distribution
corresponds to the cell center locations for a Cartesian grid type
(i_grid=1), the \(z\) point distribution contained in the grid
file for a terrain grid (i_grid=3), and a uniform point disribution
for a general mesh (i_grid=0). With the thermodynamic value sampling
points being equal to the cell centers in the Cartesian grid case, the
hydrostatic balance will be satisfied at all points within the domain.
For the other cases, however, interpolation errors, both when the
thermodynamic variables are interpolated to the cell centers, and the
errors present in interpolating the pressure to the cell faces, will
upset the hydrostatic balance. While the severity of these errors
are problem dependent, they will result in non-physical flows at early
times during the simulation since pressure and buoyance forces are out
of balance. Currently this problem can be avoided by specifying the
input i_hydrostat=1, in whinc case the conservation laws are modified by
anaylitically subtracting the hydrostatic force balance. While this
modification is straightforward for the momentum equation is is somewhat
more involved for the energy equation. For the momentum equation we
simply decompose the pressure gradient and buoyancy force terms into
background and fluctuating components and then subtract out the background
(hydrostatic) part.
\[\begin{split}\begin{eqnarray*}
\frac{\partial \rho u_i}{\partial t} & = &
-\frac{\partial}{\partial x_j}\Big( u_i \; \rho u_j \Big) +
\frac{\partial \sigma_{ij}}{\partial x_j} -
\frac{\partial p}{\partial x_i} -
\rho g\delta_{i3}\\[0.1in]
& = &
-\frac{\partial}{\partial x_j}\Big( u_i \; \rho u_j \Big) +
\frac{\partial \sigma_{ij}}{\partial x_j} -
\frac{\partial p^\prime}{\partial x_i}
\;\underbrace{ - \frac{\partial \bar{p}}{\partial x_i} -
\bar{\rho} g\delta_{i3}}_{=0} -
\rho^\prime g\delta_{i3}\\[0.1in]
& = &
-\frac{\partial}{\partial x_j}\Big( u_i \; \rho u_j \Big) +
\frac{\partial \sigma_{ij}}{\partial x_j} -
\frac{\partial p^\prime}{\partial x_i} -
\rho^\prime g\delta_{i3}
\end{eqnarray*}\end{split}\]
Here \(\sigma_{ij}\) is the viscous stress as defined in section
Governing Equations.
A similar development is used to remove the hydrostatic balance from
the energy equation.
\[\begin{split}\begin{eqnarray*}
\frac{\partial \rho E}{\partial t} & = &
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j \Big) +
\frac{\partial u_i \sigma_{ij}}{\partial x_j} -
\frac{\partial u_j p}{\partial x_j} -
w \rho g -
\frac{\partial q_j}{\partial x_j}\\[0.1in]
& = &
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j \Big) +
\frac{\partial u_i \sigma_{ij}}{\partial x_j} -
\frac{\partial u_j p^\prime}{\partial x_j} -
\frac{\partial u_j \bar{p}}{\partial x_j} -
w \bar{\rho} g -
w \rho^\prime g -
\frac{\partial q_j}{\partial x_j}\\[0.1in]
& = &
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j \Big) +
\frac{\partial u_i \sigma_{ij}}{\partial x_j} -
\frac{\partial u_j p^\prime}{\partial x_j} -
\bar{p}\frac{\partial u_j }{\partial x_j}
\;\underbrace{ - w\frac{\partial \bar{p}}{\partial z} -
w \bar{\rho} g}_{=0} -
w \rho^\prime g -
\frac{\partial q_j}{\partial x_j}\\[0.1in]
& = &
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j \Big) +
\frac{\partial u_i \sigma_{ij}}{\partial x_j} -
\frac{\partial u_j p^\prime}{\partial x_j} -
\bar{p}\frac{\partial u_j }{\partial x_j} -
w \rho^\prime g -
\frac{\partial q_j}{\partial x_j}\\[0.1in]
\end{eqnarray*}\end{split}\]
Note that an extra term, proportional to the velocity divergence, appears
in the energy equation.
Hydrostataic balance removal in conjunction with data assimilation
The data assimilation forms are derived by subtracting the background
balances from the flow equations. If the hydrostatic balance are also
to be removed we can start with the results above to derive the
corresponding data assimilation forms. Since the background equivalent
of any prime term is null, only the advective and viscous stress terms
in the momentum equation above have a background equivalent. Subtracting
these, adding the measeured momentum time rate of change, and using
the notation \(\sigma^\prime_{ij} = \sigma_{ij} - \bar{\sigma}_{ij}\),
we get
\[ \frac{\partial \rho u_i}{\partial t} =
-\frac{\partial}{\partial x_j}\Big( u_i \; \rho u_j -
\bar{u}_i \; \bar{\rho}\bar{u}_j \Big) +
\frac{\partial \sigma_{ij}^\prime}{\partial x_j} -
\frac{\partial p^\prime}{\partial x_j} -
\rho^\prime g\delta_{i3} +
\left(\frac{\partial \bar{\rho} \bar{u}_i}{\partial t}\right)_D\]
A similar procedure is used to derive the data assimilation form of the
energy equation
\[\begin{split}\frac{\partial \rho E}{\partial t} =
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j -
\bar{E} \; \bar{\rho}\bar{u}_j \Big) +
\frac{\partial}{\partial x_j}\left( u_i\sigma_{ij} -
\bar{u}_i\bar{\sigma}_{ij}\right) - \\[0.1in]
\frac{\partial u_j p^\prime}{\partial x_j} -
\bar{p}\frac{\partial u^\prime_j }{\partial x_j} -
w \rho^\prime g -
\frac{\partial q^\prime_j}{\partial x_j} +
\left(\frac{\partial \bar{\rho}\bar{E}}{\partial t}\right)_D\end{split}\]
Note that the viscous work term can be written as
\[u_i\sigma_{ij} - \bar{u}_i\bar{\sigma}_{ij} \; = \;
u_i(\sigma^\prime_{ij} + \bar{\sigma}_{ij}) - \bar{u}_i\bar{\sigma}_{ij}
\; = \;
u_i\sigma^\prime_{ij} + (u_i - \bar{u}_i)\bar{\sigma}_{ij} \; = \;
u_i\sigma^\prime_{ij} + u^\prime_i\bar{\sigma}_{ij}\]
This result may be used to rewrite the energy equation as
\[\frac{\partial \rho E}{\partial t} =
-\frac{\partial}{\partial x_j}\Big( E \; \rho u_j -
\bar{E} \; \bar{\rho}\bar{u}_j \Big) +
\frac{\partial}{\partial x_j}\Big( u_i\Sigma^\prime_{ij} +
u^\prime_i\bar{\sigma}_{ij} \Big) -
\bar{p}\frac{\partial u^\prime_j }{\partial x_j} -
w \rho^\prime g -
\frac{\partial q^\prime_j}{\partial x_j} +
\left(\frac{\partial \bar{\rho}\bar{E}}{\partial t}\right)_D\]
where \(\Sigma^\prime_{ij}=\sigma^\prime_{ij}-p^\prime\delta_{ij}\).