Suggested Development Steps for RDC Parameterization < rdctheory.cloud
[NOTE]
[NOTE]
The first half of the simplest RDC analysis described here,
without solving bothersome Poisson equations,
is sufficient to verify the presence of RDC.
This is independently summarised in
How to Easily Verify the Presence of RDC in Your Atmospheric Model
.
Please give it a try!
The first step is to estimate how much transport the RDC should carry in your current atmospheric model. The method shown here can easily estimate the RDC flow field using an external boundary value problem solver package without any modification to the atmospheric model code. For ease of analysis, we recommend a small square horizontal computational area connected with cyclic boundary conditions, for example. The parameters
must be given beforehand externally. The following RDC scheme should be applied to the region outside the supplying source domains. Time average should be taken over the time interval ΔtR.
As shown in RDC cumulus parameterization, the original form of the basic RDC equation pair, composed with the equation for the radiatively driven subsidence velocity wR and the equation of continuity, is \begin{eqnarray} \left\{\displaystyle \begin{array}{llll} {w}_{R} = {\left( \displaystyle \frac{g}{{c}_{p}} - \Gamma \right)}^{-1} {\left. \displaystyle \frac{\partial {T}}{\partial t} \right|}_{R} \hspace{6cm} & (3) & \nonumber \\ \displaystyle \frac{\partial}{\partial x} \left(\rho {u}_{R}\right) +\frac{\partial}{\partial y} \left(\rho {v}_{R}\right) = - \frac{\partial}{\partial z} \left( \rho {w}_{R} \right), & (8) & \nonumber \end{array} \right. \end{eqnarray} where all the variables appearing in the equations such as Γ, δT/δt|R, and ρ, depend on time t and space (x,y,z). When performing the simplest one-time RDC analysis for the coarsest estimate, however, it is possible to assign a time-averaged value to each of these variables. That is, the approximate system of the basic RDC equation pair shown above can be written as follows; \begin{eqnarray} \left\{\displaystyle \begin{array}{llll} {w}_{R} = {\left( \displaystyle \frac{g}{{c}_{p}} - \left<\Gamma\right> \right)}^{-1} \left< {\left. \displaystyle \frac{\partial {T}}{\partial t} \right|}_{R} \right> \hspace{5cm} & (3') & \nonumber \\ \displaystyle \frac{\partial}{\partial x} \left(\left< \rho \right> {u}_{R}\right) +\frac{\partial}{\partial y} \left(\left< \rho \right> {v}_{R}\right) = - \frac{\partial}{\partial z} \left( \left< \rho \right> {w}_{R} \right), & (8') & \nonumber \end{array} \right. \end{eqnarray} where the variables <Γ>, <δT/δt|R>, and <ρ>, are assumed to have time-averaged values over an arbitrary time interval ΔtR of, say, 10 h. This time interval is suggested to be longer than the lifetime of a single cumulus cloud and not longer than the time interval between successive cumulus cloud developments.
[NOTE]
Eqs. (3) and (8) both contain time-dependent variables such as
Γ, δT/δt|R, and ρ.
In the RDC scheme above, however, we considered net transport after time averaging.
Here, for example,
we explain why a product of two variable values can be replaced
by the product of their respective time averages.
Let's denote any two variables that make up a product as
as f(x,y,z,t) and g(x,y,z,t).
Using the symbols
< > for time-averaged parts and ' for time-varying parts,
let's divide the variable values into two parts each as
\begin{equation}
\left\{
\begin{array}{ll}
f \left( x,y,z,t \right)
& = &
\left< f \right> \left( x,y,z \right) + {f}^{\prime} \left( x,y,z,t \right)
\\
g \left( x,y,z,t \right)
& = &
\left< g \right> \left( x,y,z \right) + {g}^{\prime} \left( x,y,z,t \right),
\end{array}
\right.
\nonumber
\end{equation}
where <f>(x,y,z) and <g>(x,y,z) are the time-averages
and f'(x,y,z,t) and g'(x,y,z,t) are time-varying deviations from the averages
<f>(x,y,z) and <g>(x,y,z), respectively.
Taking time-average of the deviation parts gives zero values;
\begin{equation}
\left\{
\begin{array}{ll}
\left< {f}^{\prime} \right>
& = &
0
\\
\left< {g}^{\prime} \right>
& = &
0.
\end{array}
\right.
\nonumber
\end{equation}
Then the net effect remaining
when the product of the two variables f and g is averaged over time
is calculated as
\begin{equation}
\begin{array}{ll}
\left< f g \right>
& = &
\left<
\left( \left< f \right> + {f}^{\prime} \right)
\left( \left< g \right> + {g}^{\prime} \right)
\right>
\\
& = &
\left<
\left< f \right> \left< g \right>
+ \left< f \right> {g}^{\prime}
+ \left< g \right> {f}^{\prime}
+ {f}^{\prime} {g}^{\prime}
\right>
\\
& = &
\left< \left< f \right> \left< g \right> \right>
+ \left< \left< f \right> {g}^{\prime} \right>
+ \left< \left< g \right> {f}^{\prime} \right>
+ \left< {f}^{\prime} {g}^{\prime} \right>
\\
& = &
\left< f \right> \left< g \right>
+ \left< f \right> \left< {g}^{\prime} \right>
+ \left< g \right> \left< {f}^{\prime} \right>
+ \left< {f}^{\prime} {g}^{\prime} \right>
\\
& = &
\left< f \right> \left< g \right>
+ \left< {f}^{\prime} {g}^{\prime} \right>.
\end{array}
\nonumber
\end{equation}
The second term on the right-hand side is the time-average of
the product of time-varying parts
f'(x,y,z,t) and g'(x,y,z,t),
which is considered to be the contribution from the dynamics.
Since the physical scales of the dynamical flows are so small
that we ignore such a dynamical contribution in the RDC scheme inference
(this is the basic hypothesis in the RDC scheme),
the second term is approximately regarded to be zero;
\begin{eqnarray}
\left< {f}^{\prime} {g}^{\prime} \right>
\approx
0.
\nonumber
\end{eqnarray}
[NOTE]
When writing this page,
we assumed the relationships among physical quantities
during
the early stages of implementing the RDC scheme.
Specifically,
we assumed a Radiative-Convective model atmosphere
in a state of Radiative-Convective Equilibrium.
For this reason,
we treated time average of the product
of the disturbance terms <f'g'> as zero here.
However,
once the RDC scheme is finally incorporated into a full climate model
with an arbitrary condition of atmosphere,
the atmosphere will not necessarily be in the equilibrium state,
and <f'g'> may not be zero.
These terms will be treated dynamically
according to the equations of motion of the full climate model as usual,
separately from the RDC flow that governs the time-averaged field.
Therefore, only the first term on the right-hand side, the product of the time-averaged values <f>(x,y,z) and <g>(x,y,z), remains in the net effect after taking time-average; \begin{eqnarray} \left< f g \right> \approx \left< f \right> \left< g \right>. \nonumber \end{eqnarray}
The Poisson equation to be solved (numbered with (16) on the original page), assuming a zero-rotation RDC flow field for example, can be written corresponding to Eqs. (3') and (8') as \begin{eqnarray} \frac{{\partial}^{2} \Phi}{\partial {x}^{2}} + \frac{{\partial}^{2} \Phi}{\partial {y}^{2}} = \frac{\partial}{\partial z} \left[ \left< \rho \right> {\left( \frac{g}{{c}_{p}} - \left< \Gamma \right> \right)}^{-1} \left< {\left. \frac{\partial {T}}{\partial t} \right|}_{R} \right> \right], \hspace{1cm} & (16') & \nonumber \end{eqnarray} where Φ(x,y) is the velocity potential. Please note that this is a two-dimensional boundary value problem for Φ(x,y) on the horizontal plane (x,y) for each altitude z.
When performing the RDC analysis using the Poisson equation (16') with the time-averaged values, there is no need to modify the code of the atmospheric model itself. The model only needs to output the time averaged values over 10 hours. <Γ>, <δT/δt|R>, and <ρ>. Substituting these values into the expression in the square brackets on the right-hand-side of Eq. (16'), the value of the whole right-hand-side, the z-derivative of the square brackets, can be calculated. Now you can feed the set of the right-hand-side values into an external boundary-value problem solver package to solve the Poisson equation (16') with respect to the velocity potential Φ. Once Φ is obtained, The radiatively driven horizontal velocity field (uR,vR) is finally obtained from the Φ definition (numbered with (14) on the original page) \begin{equation} \left\{ \begin{array}{ll} {u}_{R} & = & - \frac{\displaystyle 1}{\displaystyle \left< \rho \right>} \frac{\displaystyle \partial \Phi}{\displaystyle \partial x} \\ {v}_{R} & = & - \frac{\displaystyle 1}{\displaystyle \left< \rho \right>} \frac{\displaystyle \partial \Phi}{\displaystyle \partial y}. \end{array} \right. \hspace{7cm}(14') \nonumber \end{equation} The simplest analysis shown here roughly estimates the RDC flow field uR=(uR,vR,wR) with Eqs. (3') and (14') which should occur in the model atmosphere. Even with such a rough estimate, it should be possible to show that RDC provides much more efficient transport than DD.
[NOTE]
For initial validation of the RDC,
it is recommended that atmospheric conditions be specified at low latitudes.
In mid- and high-latitude atmospheres,
mesoscale or synoptic-scale disturbances may be large enough to obscure the RDC.
Since the RDC is a universal mechanism,
it should work everywhere,
regardless of atmospheric conditions,
so that it can be verified later for mid- and high-latitudes.
You can then proceed to the next step of the RDC research, a more rigorous analysis of RDC with time-varying values of Γ, δT/δt|R, and ρ, as in the original equations (3) and (8). In this case, you can still continue to use either the cumulus-resolving model or the non-cumulus-resolving model. The Poisson equation to be solved is hence that in the original form \begin{eqnarray} \frac{{\partial}^{2} \Phi}{\partial {x}^{2}} + \frac{{\partial}^{2} \Phi}{\partial {y}^{2}} = \left< \frac{\partial}{\partial z} \left[ \rho {\left( \frac{g}{{c}_{p}} - \Gamma \right)}^{-1} {\left. \frac{\partial {T}}{\partial t} \right|}_{R} \right] \right>, \hspace{1cm} & (16'') & \nonumber \end{eqnarray} where the outmost triangular brackets on the right-hand side indicate the average over the time interval between the RDC computations. To calculate the time-mean value on the right-hand side of this equation, however, you need to incorporate the RDC calculation code into the time development code of your atmospheric model.
The final step of the RDC research will be practical use of the cumulus representation based on the RDC scheme in full climate models. Again, it would be better to start with a small horizontal area. Since the cumulus-resolving model is assumed to have the correct transport, a reliable cumulus-resolving model is used as a reference. The comparison between the results obtained from the cumulus-resolving model and the transport predicted by the RDC cumulus representation will be essential not only for rough qualitative comparisons, but also for quantitatively correct predictions of the actual atmospheric transport. The final tuning of the RDC scheme should be performed to produce the same results between these two different computations. The RDC scheme implementation would be finished considering the following points.
[NOTE]
Regarding the constraints on RDC flow
when the mass flux within a cumulus cloud is insufficient,
a logically sound solution has been found.
We can assume that a complete RDC outflow always occurs from the lower layers.
Only at the layer where the cumulus mass flux reaches its maximum altitude,
it is sufficient to impose a constant limitation on the RDC flow
over the entire reaching area.
Since the RDC scheme is based on fundamental physical principles, these tunings are not something to be performed by intuition or experience, but can be done logically and should be relatively easy.
[NOTE]
The copper-grain surface condition
employed in our 2D-DCM,
where the total energy flux is constant at any point on the surface,
may have the same effect also in the 3D atmosphere
as fixing the developing locations of the cumulus clouds.
If the locations of the cumulus clouds are fixed in the model,
the RDC-analysis would be very easy to perform.
We encourage you to give it a try.
Suggested Development Steps for RDC Parameterization < rdctheory.cloud
Exhibited on 2023/02/15
Last Updated on 2026/07/17
Copyright(C)2022-2026 jos <jos@kaleidoscheme.com> All rights reserved.