Perspectives on general coordinate models

• Motivation:– Single model (or framework/environment)

for both global scale• Adiabatic interior (hybrid coordinates)

and process studies• Non-hydrostatic

– Regional impact of global change, super-parameterization …e.g. 5-20km global resolution, 100m nested regional resolution (“Mosaics”)This will be a reality within the decade (or a few years).

• Can a Lagrangian (layered) class ocean model include non-hydrostatic effects?

• Pertinent issues originally noted by Bleck, Schopf and others– Recently discussed in note:

“On methods for solving the oceanic equations of motion in general coordinates”, Adcroft and Hallberg (2005), Ocean Modell. 8 (?)

with which we hope to re-invigorate the discussion.

Hydrostatic (Boussinesq) equations in height coordinates: “z”

• 7 unknowns, 4 prognostic eqns, 3 diagnostic eqns

• 2 x Gravity mode• 1 x Planetary mode• 1 x thermo-haline mode

– Free-surface equation obtained from continuity + B.C.s

p)θ,ρ(s,ρQswvssQθwvθθ0 w v 0pρ

F pv2ΩvD

szhzt

θzhzt

zhz

z

zρ1

hht o

g

ρsθp

Dw0) v,(u,vh

zt

Hydrostatic (Boussinesq) equations in isopycnal coordinates: “ρ”

• 8 unknowns, 7 equations– 5 prognostic eqns, 2 diagnostic eqns

– 8th equation: prescribe

p)θ,α(s,α

Qρszvszsz

Qρθzvθzθz

0ρzvzz

0pαgz

Fgzpαv2ΩvD

sρρhρρρt

θρrhρρρt

ρρhρρρt

ρρ

ρρhht

αsθ

zp

0) v,(u,v

ρ

h

ρQρ

ρ

Hydrostatic (Boussinesq) equations in general coordinates: “r”

• Coordinate transformation:• is “thickness”• 8 unknowns, 7 equations

8th equation? or

p)θ,ρ(s,ρQrszvszszQrθzvθzθz0rzvzz0pgzρ

Fgzpv2ΩvD

srrhrrrt

θrrhrrrt

rrhrrrt

rr

rρρ

rρ1

hht oo

r

h

zρsθpr

0) v,(u,v

rQr )H,η,z(r,z

t)z,y,r(x,r

zzxytr r

Using the continuity equation

Eulerian VerticalDynamics Method (EVD)

• Specifies

• Uses continuity diagnostically to find

Lagrangian VerticalDynamics Method (LVD)

• Specifies

(Inconsistent with a N-H vertical momentum equation?)

• Uses continuity to predict

0rzvzz rrhrrrt

)Hr,η,,η(z rt f

rz

hrrrr vzrz f

rrQr

rrrhrrrt Qzvzz

Unlikely to recover “adiabatic” properties of isopycnal models

Non-hydrostatic (Boussinesq) equations in height coordinates (z)

Momentum (3d)

Continuity (Volume)

Temperature, salt and E.O.S.

• Seven degrees of freedom– u,v,w,ρ,θ,s,p

• Seven equations– 5 prognostic + 2 relations– No eqns for p

0ˆ1

00

kgpKvt

0 v

p)θ,ρ(s,ρ QsD QθD stθt

Solving the non-hydrostatic equations in height coordinates: “projection method”

Momentum (3d)

Continuity (Volume)

Essential Algorithm

21

21 ˆ11

00

1

nnnn GkgKpvvt

010 nv

21

21

21

0

1

0

*

*

*

nn

n

nn

ptvv

vpttGvv

Constraint on flow= Equation for pressure!

Projection method in LVD mode?Momentum (3d) (as before)

Continuity

Using the Eulerian approach:

21

21 ˆ11

00

1

nnnn GkgKpvvt

0rzvzzzΔt1 1n

rr1

hnrr

nr

*r nn

If this is prescribed we can not insert the vertical momentum equation here

ρDrHDrηDrzz

przppz

tρtHtηrr*rΔt

ρ

rzrrrrrrr

0

Arbitrary Lagrangian-Eulerian method (ALE)?

• Lagrangian phase

• (Optional) Eulerian phase (remapping)

0vzzzΔt1

hnrr

nr

*r

)z(z *r

1nr M 0rzzz

Δt1 1/n*

rr*r

1nr

To make this N-H, we have to already know the flow by this point.

The EVD approach tries to constrain the N-H pressure with the final form of continuity

Hydrostatic/non-hydrostatic decomposition

• Decompose pressure into parts– Surface (ps)

– Hydrostatic (ph)

– Non-hydrostatic (pnh)

00

0

1

0

1

21

21

21

21

21

21

21

1

ˆ11

11

gp

GkKpwwt

GKpppvvt

nhz

nwz

nnhz

nn

nhhh

nnh

nh

ns

nh

nh

Non-hydrostatic mode• 2D + 3D elliptic problem

21

21

21

21

21

21

21

21

0

1

0

22

0

0

10

0

**

****

***

***

*

*

nnhhh

nh

wzwzzhhnnhz

nnh

wnw

n

nshhh

n

kkh

nshh

nhh

nh

nhh

ptvv

GtGtwvppttGwtGww

ptvv

vzpHt

pttGvv

k

Hydr

osta

tic Can use EVD or LVD up until this point

N-H

upda

te

Non-hydrostatic modeling in general coordinates

• Explicit solution of Navier-Stokes equations– Continuity leads to a prognostic equation for

pressure

– Can be integrated in any coordinate system– Separation of time scales in ocean is prohibitive

sT QQ vρpDc1

3t2s

TC

QpvC

pCpD

vv

pt

Ocean Atmosphere

1-igw

-1s

s m 2c

s m 1500c

1-

-1s

s m 50U

s m 300c

Ocean Atmosphere

Points to take home

• Some hybrid coordinate models use Eulerian paradigm (not HyCOM, HyPOP, Poseidon)

– need to assess adiabaticity• Lagrangian paradigm

– Easy to make adiabatic– Harder to make non-hydrostatic (not impossible)– Breaks symmetry between horizontal and vertical

directions