Optimal Boundary Conditions for ORCA2 model.
A 4DVar data assimilation technique is applied to a ORCA2 configuration of the NEMO in order to identify the optimal parametrization of the boundary conditions on the lateral boundaries as well as on the bottom and on the surface of the ocean. The influence of the boundary conditions on the solution is analyzed as in the assimilation window and beyond the window.
It is shown that optimal conditions for vertical operators allows to get stronger and finer jet streams (Gulf Stream, Kuroshio) in the solution. Analyzing the reasons of the jets reinforcement, we see that the major impact of the data assimilation is made on the parametrization of the bottom boundary conditions for lateral velocities u and v.
We modify discretizations of vertical operators near boundaries (surface and bottom)
and look for the optimal discretization by variational data assimilation
The vertical model grid is organised as follows:
The index k=0 corresponds to the surface, k=K to the bottom. All variables are devined at T levels, except the vertical velocity w.
To calculate interpolations and derivatives at levels, adjacent to the surface or to the bottom, we use the control coefficients α for all interpolations and derivatives in the following way:
Interpolation of the w to T levels
w_{½}=α_{0}^{St}+
α_{1}^{St}w_{0}+
α_{2}^{St}w_{1},
w_{k+½}= 1/2(w_{k}+w_{k+1}),
w_{K½}=α_{0}^{Sb}+
α_{1}^{Sb}w_{K}+
α_{2}^{Sb}w_{K1}
Derivative of the T at w levels
(dT/dz)_{1}=α_{0}^{Dt}+
α_{1}^{Dt}T_{½}+
α_{2}^{St}T_{½},
(dT/dz)_{k}= 1/h(T_{k+½}T_{k½}),
(dT/dz)_{K1}=α_{0}^{Db}+
α_{1}^{Db}T_{K½}+
α_{2}^{Sb}w_{K1½}
Moreover, we add control coefficients to the calculation of the vertical velocity from horizontal velocity divergence
and to the vertical dissipation/diffusion operators
Thus, operators are allowed to change their properties near boundaries in order to find the best fit with requirements of the model and data. To assign all control variables α we perform data assimilation procedure and find their optimal values.
Observational data are assimilated into the model during 10 days interval, from the 1st to 10th of January, 2006. Examining the model solution on the 30 days interval, from the 1st to 30th of January, 2006, we get
Stronger and Finer Gulf Stream
Click on the images to see the evolution of the Sea Surface Elevation in the North Atlantic from the 1st to the 30th of January 2006.
(animated gif files, about 10M)

Optimal Boundary Conditions for vertical operators

Optimal Initial Conditions



Stronger and Finer Kuroshio
Click on the images to see the evolution of the Sea Surface Elevation in the North Pacific from the 1st to the 30th of January 2006.
(animated gif files, about 8M)

Optimal Boundary Conditions for vertical operators

Optimal Initial Conditions



The Question: Modification of which boundary conditions results in the streams reinforcement?
Sea surface elevation is directly related to the vertical velocity on the surface. We control w on the surface adding control coefficients α_{0}^{ws} and α_{1}^{ws} (see formula). So far, much observational data is collected by satellites and available for the sea surface elevation, this variable might undergo a strong forcing pulling it toward observations. It is possible, consequently, that data assimilation modifies just coefficients α^{ws} on the surface for the vertical velocity and this modification is immediately translated to SSH with no influence on all other variables.
To verify this hypothesis, we plot α^{ws} on the surface and α^{wb} on the bottom
Control coefficients α^{w} for the vertical velocity in the North Atlantic

α_{0}^{ws} on the surface

α_{0}^{wb} on the bottom



α_{1}^{ws} on the surface

α_{1}^{wb} on the bottom



Conclusion :
 The magnitude of α_{0}^{ws} is about 10^{7} while the vertical velocity on the surface exceeds 10^{5};
 No characteristical stream pattern can be observed in the pictures of α^{ws} on the surface;
 Control coefficients α_{1}^{w} are bigger on the bottom than on the surface
Examining the Vertical velocity itself
Vertical velocity, surface of the North Atlantic, January, 30, 2006.

Original model, surface

Optimal vertical BC, surface



Vertical velocity, xz and yz section North Atlantic, January, 30, 2006.

Original model, yz section

Optimal vertical BC, yz section



Original model, xz section

Optimal vertical BC, xz section



Conclusion :
 Very small modification of the vertical velocity on the surface
 Strong modification of the vertical velocity near the bottom
A question we can ask :
whether coefficients α^{w} create fountains on the bottom that spray out jets of water, or they create threedimensional eddies in which the divergence of lateral velocities is balanced by ∂w/∂z?
Threedimensional vortex near the bottom (K1,K2 levels)

Velocity u Level K2

Velocity v Level K2

Velocity w Level K2




Velocity u Level K1

Velocity v Level K1

Velocity w Level K1




Conclusion :
 Big w near the bottom must be due to modifications of the lateral velocity components.
Examining the control of lateral velocities
we look at the vertical diffusion operator (see formula)
Control coefficients α^{Dzz} for the vertical diffusion operator in the North Atlantic

α_{0}^{Dzzs} on the surface

α_{0}^{Dzzb} on the bottom



Conclusion :
 Very little control in the jetstream area on the surface
 Strong control in the jetstream area on the bottom
Examining the lateral velocities themselves
Velocity u, yz section North Atlantic, January, 30, 2006.

Original model, yz section

Optimal vertical BC, yz section



Velocity v, yz section North Atlantic, January, 30, 2006.

Original model, yz section

Optimal vertical BC, yz section



Conclusion :
The positive anomaly of u just below the jet stream (39^{o}...41^{o}N) maintains the flux over the whole water column ensuring more than 10 cm/s velocity at the 500 m depth. That means the Gulf Stream become not only stronger (one can see the surface velocity exceeds 35 cm/s instead of 25 cm/s on the left) and longer, but also deeper.
This fact brings us to the conclusion that the major impact of the data assimilation is made on the parametrization bottom boundary conditions for lateral velocities. Consequently, the modification of the sea surface elevation in the jetstream regions is not a simple artefact, but the result of optimization of the deep dynamics.
For more information,
see
E. Kazantsev.
Optimal Boundary Conditions for ORCA2 model.
submitted to Ocean Dynamics
Eugene.Kazantsev@imag.fr
Last modified the 11th of December 2012