Optimal Boundary Conditions for ORCA2 model.

A 4D-Var data assimilation technique is applied to a ORCA-2 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 w½0St+ α1Stw0+ α2Stw1, wk+½= 1/2(wk+wk+1), wK-½0Sb+ α1SbwK+ α2SbwK-1
Derivative of the T at w levels (dT/dz)10Dt+ α1DtT½+ α2StT½, (dT/dz)k= 1/h(Tk+½-Tk-½), (dT/dz)K-10Db+ α1DbTK-½+ α2SbwK-1½
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 α0ws and α1ws (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 αws 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 αws on the surface and αwb on the bottom
 Control coefficients αw for the vertical velocity in the North Atlantic α0ws on the surface α0wb on the bottom α1ws on the surface α1wb on the bottom
Conclusion :
• The magnitude of α0ws 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 αws on the surface;
• Control coefficients α1w 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, x-z and y-z section North Atlantic, January, 30, 2006. Original model, y-z section Optimal vertical BC, y-z section Original model, x-z section Optimal vertical BC, x-z 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 αw create fountains on the bottom that spray out jets of water, or they create three-dimensional eddies in which the divergence of lateral velocities is balanced by ∂w/∂z?
 Three-dimensional vortex near the bottom (K-1,K-2 levels) Velocity u Level K-2 Velocity v Level K-2 Velocity w Level K-2 Velocity u Level K-1 Velocity v Level K-1 Velocity w Level K-1
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 αDzz for the vertical diffusion operator in the North Atlantic α0Dzz-s on the surface α0Dzz-b on the bottom
Conclusion :
• Very little control in the jet-stream area on the surface
• Strong control in the jet-stream area on the bottom
Examining the lateral velocities themselves
 Velocity u, y-z section North Atlantic, January, 30, 2006. Original model, y-z section Optimal vertical BC, y-z section Velocity v, y-z section North Atlantic, January, 30, 2006. Original model, y-z section Optimal vertical BC, y-z section
Conclusion : The positive anomaly of u just below the jet stream (39o...41oN) 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 jet-stream regions is not a simple artefact, but the result of optimization of the deep dynamics.