Ocean Basins
We now want to consider instability, transition to t~irb~ilence:
and t~irb~ilence
in an
ocean basin. There are lots of systems t,o choose frorn. The Stornrnel rnodel appears to be
stable even a t strong forcing.
q = v2g
For these, hf = 0.032.
Demos, Page 1: S t o m m e l 1 g y r e
< l i n e a r model>
<Stomme1 di=O. 023>
<zeta>
<di=O. 071>
<zeta>
<di=O. 23>
<zeta>
<zeta>
Demos, Page 1 :
<di=O. 093>
<q>
Two g y r e
<di=O. 0093>
<q>
<di=O. 030>
<q>
The Mlink model
is livelier.
Demos, Page 1 : Munk <di=O. 0037> <di=O. 0053> <di=O. 0064> <di=O. 0074>
<di=O. 0083>
<di=O. 010>
<di=O. 012>
<summary>
Demos, Page 1 : Munk t w o g y r e <di=O. 0037> <di=O. 0053> <di=O. 0064>
<di=O. 0074>
<di=O. 0083>
<di=O. 010>
<di=O. 012>
<summary>
Stability and Bifurcations
Steady states
Cessi and Ierley (.JPO, 25, 1196-1205) fo~indsteady states by discretizing (nsing
Chebyshev polynomials) and iising Newton's method t,o find conditions where $q = 0.
Some soliltions can be fo~indby starting with the visco~issohition and gradiially shifting
the paramet,ers; ohters require initial guesses. Once fo~ind,t,hese can then be continlied.
Markings indicate mirnber of steady states fo~ind;
(1) one antisymmetric
(3) one anti + 2 non (5a) one anti 4 non 1 non (5b) three anti
Demos, Page 1 : R e g i m e s C e s s i and I e r l e y < o v e r a l l : xv - g e o m e t r y -0+0
-glenn/12.822t/graphics/cil.j p g > closnipci2.jpg < d f = 4 e - 2 , di=2e-2>
<di=3e-
+
+
2>
<di=4e-2>
<A2 and A3>
<di=5e-2>
<di=6e-2>
<summary>
<A1 and N1>
<N2>
The experiments shown have hf = 4 x 10V2 and 6,i = 1 . .. 6 x l o p 2 so that they cross the
regimes shown.
Bifiircations and stability
The bifurcation sequence can be inferred by looking a t where the roots coalesce;
stability can be foiind by discretizing the linearize equations and looking for eigenvaliies
with posit,ive real parts. Alternatively one can integrate the linearized equations start,ing
growing modes to appear.
with random perturbations and look for exp~nent~ially
2:
Demos, Page 2 : b i f u r c a t i o n s
stability
<regions>
<df=0.075>
<df=0.0478>
Demos, Page
Baroclinic case
The two-layer problem allows for baroclinic instability as well, with significant changes
in the dynamics. In principle the two-layer system allows an upper-layer only sohition:
like the Miink soliit,ion biit with the addition of a deformation radius to the PV. If the
diffiision acts only on the relative vorticity, the RCit erm woiild drop out in steady state; in
our model, the diffiisivity acts on the PV, so it woiild still appear, But the spin-lip to this
stat,e is complicat,ed. We tend to think of each mode independently in the linear problem:
so that the barotropic wave crosses the basin, followed rnlich more slowly by the baroclinic
wave which wipes oiit the deep signal; leaving the Sverdr~ipflow in the upper layer only.
But, this picture does not acco~intfor the boiindary conditions. Cane and Sarachik (1976:
JMR,, 34, 629-665) showed that the spin-lip involves Kelvin waves crossing to the east along
the eqliator or souther boiindary, moving lip the eastern wall, and generating Rossby waves
which then cross the basin to the west.
The QG model has a representation of this process in the baroclinic bo~indarycondition. If we take the transformed geostrophic m o n ~ m t , ~ i interface,
m,
and mass eqiiations
and not,e that the transformed variables still satisfy ut . n = 0 on the walls, we find
The mass of each layer rnnst be conserved. In addition, this implies that the circ~ilation
aroiind the basin is altered only by direct forcing and dissipation. From these statements,
we can alignlent the ?i,3- - r:3 -(t) condit,ions with a forc~ilafor cl cz (the barotropic part
does not matt,er). Alternat~vely,when we invert the PV, we add snfficient free soliltion
-
so that the int,egrals of
are zero.
Demos, Page 2 : bc spinup <correction field>
We can avoid this problem by iising the double-gyre.
<no corr>
<corr> Berloff and Meacharn (JPO, 28, 361-388) have explored the two-layer; single-gyre
version of the same problem.
Demos, Page 3 : Berloff and Meacham
<bifurcation>
<limit cycle> <more complex>
<chaos> We can see the transit,ions in the two-layer doiible gyre model. By comparison, the
case with infinitely deep lower layer has steady sohitions
Demos, Page 3 : Transition
<tau 0.003>
<tau 0.004>
<tau 0.005> <tau 0.004 del=O>
<q>
<q tot> Demos, Page 3 : Example <single layer>
<two layer>
<ql>
<qtl> <qt2>
<psil>
<psi2>
<h>