Fundamental equations and methods

1
Fundamental equations and methods

• This session is aimed at establishing a
  foundation for fundamental issues involved with
  building an ocean model. The following points
  will be addressed in this session
• Physical assumptions and resulting mathematical
  equations discretized in an ocean model
• Boussinesq versus non-Boussinesq
• Hydrostatic versus non-hydrostatic
• Finite differences, finite volume, and finite
  element methods (Iskandarani)
• Horizontal and vertical meshes Sessions 2 3
• Algorithms for Eulerian and Lagrangian vertical
  coordinates (Dukowicz)
• Algorithms/time stepping for fast and slow modes
• Advective form momentum versus vector invariant
  form velocity
• Calculation of pressure forces
• Advective tracer transport
• Fundamental principles that an ideal ocean model
  should respect (e.g., conservation,
  realizability, etc.)
• Too much material to do justice to in 1/2 hour
  (or even 1½ hrs)
• Will focus on some outstanding questions and/or
  curiosities

2
The equations we cant afford to solve

• Navier-Stokes equations
• Cant solve for climate for two reasons
• Scales at which molecular processes act are tiny
  (Kolmogorov )
• Permits sounds waves

Usually replace T with ?
3
1. Reynolds averaging

• Define some average/filter/ensemble s.t.
• Minefield for the non-theoretical!
• Note absence of Reynolds fluxes in continuity
• Due to definition of flow as rate of mass
  transport?
• Molecular terms ltlt eddy terms
• Can we drop molecular terms?
• Curiosity - B.C.s for eddy fluxes?

4
2. Muting the ocean (filtering sound)

• Sound waves essentially linear
• Very fast
• Filter with either
• An-elastic approximation
• Hydrostatic balance (vertical momentum)
• Lamb mode (where is it?)
• (often use both) (or modifying E.O.S.) (or
  )

5
Scaling continuity

• Partition the full flow into
• horizontally non-divergence component
• horizontally divergent/3D non-divergent
• 3D divergent component

6
Oceanic Boussinesq approximation

• Boussinesq approximation is most associated with
  z-coordinates for historical reasons
• rigid-lid -gt z-coordinates -gt Boussinesq
• Would not consider it now if we hadnt already
  used it!
• Justification
• Misunderstandings
• because

7
Isomorphisms

• Boussinesq (z)
• Free boundary
• Rigid boundary

• Non-Boussinesq (p)
• Free boundary
• Rigid boundary

Identical dynamics described by
Boussinesq/non-Boussinesq equations
8
Energy (currency of physics)

• Navier-Stokes energetics
• Conversion between K?F and K?I
• Boussinesq energetics
• This is what the real fluid would be doing with
  the Boussinesq solution!
• Do models need to conserve K, ? and I?
• and has any model ever done so?
• Conserve versus diminish?

9
Non-hydrostatic modeling

• Scaling the vertical momentum equation
• Non-dimensional
• Ratios
• Note that Marshall et al., 95 obtained from the
  buoyancy equation
• For large scale flows, makes
  smaller
• NH modeling is more expensive
• NH modeling is easiest with Boussinesq

10
Assumptions unworthy of mention

• Thin atmosphere (ocean) approximation
• Trivial to relax by modifying geometry of grid
• Spherical earth
• Easy to include, again through geometric terms
• Constant gravitational acceleration
• Only if you use g!
• Neglect of self-attraction and loading
• Tough one but needed for tides. Not climate?
• Circulation alteration by ocean fauna
• Surface tension
• Some rocks are more solid than others
• Two component fluid (solution energy of CFCs!)
• Non-continuum nature of a fluid (Brownian motion)
• Relativistic corrections

11
Properties of the continuum to guide numerics

• Tracer equations exhibit various symmetries
• Higher moments (used by Bryan, 1969)
• Led to particular spatial discretizations
  (ignoring time)
• e.g. globally conserve ?2
• Higher accuracy regains some credibility
• Observing monotonicity/extrema
• Has led to flux correctors/limiters
• Space-time treatment

Unlimited schemes of various orders
As above with various limiters
Which is better physically more realizable
solution or formally more accurate?
12
Vector invariant form

• Traditionally z-coordinate models used
  conservative form of momentum equations
• Requires metric terms specific to horizontal
  coordinate system
• Layered models more often than not use the vector
  invariant form
• Invariant for orthogonal coordinate systems
• Very similar form for non-orthogonal coordinates
• Very convenient to access PV Q(f?)/h
• and PV related quantities (hQ, Q2, etc)
• And KE

13
Similar approaches for momentum?

• PV is a scalar (PV eqn is a tracer equation)
• In layered framework, Coriolis terms are just
  the advective transport of PV
• Applying tracer technology to q avoids
  oscillations in q
• If we can do same for K, then can drop explicit
  viscosity needed for numerical stability
• By advecting PV, flow may change
• Q Do passive tracer principles apply to PV?

14
Burgers equation

• Inviscid self-advection of momentum
• Centered F.D. can be very unstable
• Non-linear methods (e.g. upwinding) produces
  physically more realizable solutions
• Enquist-Oscilles scheme
• Gudonov scheme
• Numerical viscosity versus explicit dissipation?
• Which is better physically more realizable
  solution or formally more accurate?

15
Solvers!

• Traditionally, ocean models solve each equation
  independently
• CFD solvers often solve system as a whole
• With all the limiters etc
• Any advantage of one approach over the other?