Inviscid Dissipation Catalyzed by Topography

INI, Oct. 18, 2010

WK DewarFSU

Collaborators: Prof. P. Berloff, Prof. A. Mc. Hogg, Prof. JC McWilliams

Work offered in the memory of PD Killworth

An analysis of ocean power, after Wunsch and Ferrari, 2004.

And there is a related potential vorticity budget question.

Connection to themeof School: Are theseprocesses that can berepresented as a stochastic process?

Numerical Evidence in Support

An ultra-fine embedded solution for Monterey Bay

(Simpler)NumericalEvidence inSupport

Temperature at western wall

Next obvious question

Experiment used vertical walls, which don’t exist, really, in the open ocean.

What happens for, say, a seamount.

Some danger here, as topography is difficult to represent numerically.

Summary:

Something very interesting happens at topography when balanced flow interactions with it.

Question:

What the hell is it?

Focus of the rest of the talk.

Theory of Wall Interactions

ut uux vuy fv M x

vt uvx vvy fu M y

M p gz

qt uqx vqy 0; q ( f vx uy ) / z

EOMs in density coordinates

zt (uz )x (vz )y 0

At the wall, normal flow vanishesand consider inviscid, adiabatic flow

u=0

M xx f2M

M 0

M gz

Decompose v into geostrophic imprint and wall response

M 'xx f2M 'M

0

SETEICST

Hydrostatic?

fv ' M x

v 'tv '2

2

y

vgv ' yM 'y vgt vgvgy M gy

Solutions:

MITgcm

Anticyclone/Cyclone Asymmetry captured, as are amplitudes, density overturnings, rates of development

M xx f2M

M 0

fv M x

vt v2

2

y

(vgv)y M y vgt vgvgy M gy

M nt (vg fRdn )M ny vgyM n

V

Asymmetry?

Source of Instability?• Linearize and

M xx f2M

M 0

fv M x

vt vgvy M y 0

vg vg ()

(vg c)M xŽ f MŽ 0

(vg c)2MŽ M MŽ 0

(vg cr )| (vg c)2 |2

|MŽ |2 d 0

Instability is not baroclinic

Source of Instability?• Linearize and

M xx f2M

M 0

fv M x

vt vgvy M y vgyv

vg vg (y)

M nt (vg fRdn )M ny vgyM n

V

Y

What does nonlinearity do?go to reduced gravity frame

M xx f2 M

Rd0

fv M x

vt v2

2

y

(vgv)y M y vgt vgvgy M gy

!

Temperature at western wall

4. How fast does the shock go?

csvy v2

2

y

(vgv)y M y (F(vg ))y

cs

M

M

cs M M

2 fR vg c fR

Non-hydrostatic dynamics arrest the shock

(zq)t (uqz )x vqz y(Yx Xy ) (uH )y (vH )x

uzq 'dyd vH o dyd O( fR)2

(Simpler)NumericalEvidence inSupport

Summary and to do list

Simple, surprisingly accurate local theory for wall interactions can be written

• predict anisotropy• steepening and breaking• can be parameterized• stochastic parameterization?

Extend this to non-wall topography?(Yes?)

Introduce into models to control boundary dissipation