Flow-Topography Interaction

Valsavarenche Summer School

June 16, 2014

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

POINT: The ‘slow’ winds input about 0.9 TW to the large-scale, balanced, oceanic circulation. From there, instabilities energize mesoscale. - As near as can be told, the mesoscale appears not to be accelerating, so 0.9TW leave the mesoscale. QUESTION: How does it get out? (I will argue topography appears to provide a potentially important pathway.)

H

L

What about external effects like boundaries?

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.

Really don’t understand what happens with realistic topography. Mr. Bishnu has been attempting to explain this to me. And, as you can imagine, this is not easy to do.

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 + Hu! " fv = "Mx + X

vt + uvx + vvy + Hv! + fu = "My +Y

M = p + !gzqt + uqx + vqy = 0; q = ( f + vx ! uy ) / z"

EOMs in density coordinates

M ! = gz

z!t + (uz! )x + (vz! )y + Hz! = 0

ut + uux + vuy ! fv = !Mx

vt + uvx + vvy + fu = !My

M ! = gzM = p + !gz

qt + uqx + vqy = 0; q = ( f + vx ! uy ) / z"

At the wall, normal flow vanishes and consider inviscid, adiabatic flow

u=0 q ! 0

f (vg + v ') = Mgx + M 'x

v 't+v '2

2!"#

$%& y

+ (vgv ')y + M 'y = 'vgt ' vgvgy ' Mgy

Decompose v into geostrophic imprint and wall response

These equations are only subject to the Hydrostatic approximation on the wall. Also, Lagrangian solution of pv equation is correct to the same approximation

q(x, y,!,t) = q(xo, yo,!,0); x = xo + udt; y = yo + vdt0

t

"0

t

"

qt + uqx + vqy = 0; q = ( f + vx ! uy ) / z"

u=0 q ! 0

Wall response has no pv anomaly

vx ! uy + fM ""

=f

M ""# vx ! uy =

fM ""

M '""

First (and only) ‘real approximation’

ut + uux + vuy! fv = !Mx

vx ! uy= "

These entail errors of O(U/V)<<1 • kinematically enforced by the wall • accurate to hydrostatic approximation on the wall

fv = Mx

vx = fM '!!M !!

Mxx " f 2M !!

M !!= 0

Consequences:

a linear elliptic equation! seteicsi

Mxx ! f 2M ""

M ""= 0

fv = Mx

vt +v2

2!"#

$%& y

+ (vgv)y + My = 'vgt ' vgvgy ' Mgy

Needs either Neumann or Dirichlet boundary conditions for complete solution

A complete, simple, remarkably accurate theory of the wall

Now, it’s time for the four most important words in science

I Don’t Believe you.

More precisely, does the theory work? We have tested it against MITgcm solutions.

Solutions:

Y Km

!

Day 0

40 80 120 160 200

0.3

0.225

0.15

0.075

0

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Y Km

!

Day 0

40 80 120 160 200

0.3

0.225

0.15

0.075

0

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Y Km

Day 4

Total Velocity, Northward Moving Anticyclone

Total Velocity, Southward Moving Cyclone

40 80 120 160 200

0.3

0.225

0.15

0.075

0

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Y Km

Day 4

CI=0.02m/s

40 80 120 160 200

0.3

0.225

0.15

0.075

0

−0.2

−0.1

0

0.1

0.2

0.3

0.4 MITgcm

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

Source of Instability and asymmetry? •  Linearize and

Mxx ! f 2M ""

M ""= 0

fv = Mx

vt + vgvy + My = !vgyv

vg = vg (y)

Mnt + (vg ! fRdn )Mny = !vgyMn

V

Y

What does nonlinearity do? . go to reduced gravity frame

Mxx ! f 2MRd

= 0

fv = Mx

vt +v2

2!"#

$%& y

+ (vgv)y + My = 'vgt ' vgvgy ' Mgy

!

Temperature at western wall

How fast does the shock go?

!csvy +v2

2"#$

%&' y

+ (vgv)y + My = (F(vg ))y

cs

M !

M +

cs =M + + M !

2 fR+ vg ! c ! fR

Non-hydrostatic dynamics arrest the shock

(z!q)t + (uqz! )x + vqz!( )y = (Yx " Xy ) + (u!H )y " (v!H )x

uz!q 'dyd! = v!H"# $%o dyd!&&&& = O( fR)2

Summary and to do list Simple, surprisingly accurate local theory for wall interactions can be written

•  predict anisotropy •  steepening and breaking •  can be parameterized

Extend this to non-wall topography? (Yes)? Introduce into models to control boundary dissipation

Numerical Evidence in Support

An ultra-fine embedded solution for Monterey Bay