Kinetic Energy Equation Boussinesq

1
Kinetic Energy Equation(Boussinesq)

• Momentum equations
• Kinetic energy equation
• Following relations need to be satisfied by model

2
KE Conservation Coriolis Term

• Arakawa C grid staggers velocity components in
  space
• Need to satisfy
• Says that u(-fv) from x-equation
• balances v( fu) from y-equation

In u-equation at (i-½,j)
In v-equation at (i,j-½)
3
Energy conservation and time-stepping I

• Evolution of first and second moments
• Discretized explicitly-in-time as
• If uf(u) satisfies some constraints then the
  discrete expression should also
• Very difficult in the above form
• Only scheme capable of exact conservation is TI
  (C-N)

4
Energy conservation and time-stepping II
5
Energy conservation and time-stepping III
6
Dissipation and damping

• Damping due to filtering does not amount to
  dissipation
• Steady state of
• is
• and is recovered by filtered models.
• If a scheme adds dissipation, such as
• then steady state is different
• Only transients are damped by filtering!

7
Summary of Numerical Issuesfor Energy
Conservation

• Spatial discretization cannot conserve energy
  conservation except when using a perfect
  time-derivative or solution is steady
• Using an accurate integrator greatly improves
  situation
• Steady state diagnostics

8
Potential Energy Conundrum

• Potential Energy Equation
• Boussinesq model does not conserve mass
• Using a linear EOS
• Arbitrary reference level ? z?z-zo
• Both Boussinesq approx and ??gz at fault

9
Energy in Boussinesq Approx.

• Non-Boussinesq Boussinesq