SOES6002 Module A, part 2: Multiple equilibria in the THC system: Governing equation Tim Henstock.

SOES6002 Module A, part 2:Multiple equilibria in the THC

system:Governing equation

Tim Henstock

What is the objective?

• Use a simplified model to analyse stability of Thermohaline circulation system

• Variety of approaches:– Analytical solution of simple system– Determination of equilibrium solutions– Non-dimensional method (suggests intrinsic scale

for problem)– Stability analysis for equilibria

• Shows possible strategies as well as this specific case

Box model

• Stommel (1961), Marotzke (1990)

Key aspects

• Parameters– T1, T2 temperature in each box

– S1, S2 salinity in each box

– q strength of circulation

• Assume that atmosphere controls:– T1, T2

– HS virtual salinity flux due to evaporation/ precipitation balance E, HS=S0E/D

Dynamics

• Flow law for q:

(2)

• Equation of state

(3)

• ie

(5)

210

k

q

iii ST 10

1212 SSTTkq

Conservation equations

• Temperature imposed externally, so no need to consider heat

• Salt:

(6)

(7)

• Note that box 1 imports S2 and exports S1, independent of the sign of q

• Also (6)+(7)=0, ie total salt is conserved

121 SSqHS s

122 SSqHS s

New approach

• Consider north-south (meridional) differences in the properties, and recast equations:

(8)

(NB deliberately different senses!)

so

(9)

211222 ;; TTTSSS

STkk

q

0

Final equation

• Recast (6), (7) in terms of S

(13)

• So, using (9)

(14)

• Entire behaviour of model controlled by (14)

SqHSSS S 2212

SSTkHS S 22

Summary

• 2-box model for thermohaline circulation

• Temperature imposed by atmosphere

• Simple dynamics, controlled by density differences

• Behaviour described by single equation

SSTkHS S 22


system:Equilibrium solutions

Tim Henstock

Recap and plan

• 2-box model for thermohaline circulation

• Used dynamics and conservation equations to obtain:

(14)

• Next, look for equilibrium solutions

SSTkHS S 22

Application

• Usually net evaporation near equator (warm), net precipitation near pole (cold), so temperature, salinity both higher near equator than poles

• Look for equilibrium (steady-state) solutions for S

(15)

• Distinguish two cases:

0 SSTkH S

Case 1 (1):

• Effect of T more important than S, so polar density higher

• Surface flow poleward, q>0

(10)

• T drives circulation, S brakes

(16)

STkqq

STq ;0

Case 1 (2):

• Hence

(17)

• or

(18)

• With solutions:

(19)

0 SSTkH S

02

kHTSS S

22,1 4

1

2

1

Tk

HTS S

Case 1 (3):

• Real solutions need positive radicand

(20)

• Two equilibrium solutions with poleward flow– Thermally dominated/thermally direct

• If freshwater forcing exceeds threshold of (20), no thermally-driven equilibrium

4

12

Tk

H S

Case 2 (1):

• Effect of S more important than T, so polar density lower

• Surface flow equatorward, q<0

(11)

• S drives circulation, T brakes

(21)

TSkqq

STq ;0

Case 2 (2):

• Hence

(22)

• or

(23)

• With solution:

(24)

0 SSTkH S

02

kHTSS S

23 4

1

2

1

Tk

HTS S

Case 2 (3):

• Must discard negative solution, otherwise contradict (21)

• Single equilibrium solution with equatorward flow– Salinity dominated/thermally indirect

• Salinity driven equilibrium exists for all positive values of freshwater forcing

Equilibrium solutions (1):

• Define dimensionless parameters:

Salinity difference:

Salinity flux:

• Note δ<1 implies q>0, ie poleward flow, salinity driving force relative to external temperature driving force

TS

2TkHE S

Equilibrium solutions (2)

Equator-ward flow

Pole-ward flow


• Three steady state solutions if freshwater flux not too strong, ie

• Two have poleward flow– Small salinity contrast, strong flow δ<0.5– Large salinity contrast, weak flow δ>0.5

• One has equatorward flow– Large salinity contrast, always exists

25.02 Tk

HE S

Equilibrium solutions (4)

Equator-ward flow

Pole-ward flow

Weak flow

Strong flow


• Poleward (thermally direct) solution disappears if

• To overcome surface salinity flux must increase salinity advection qS:– Increase salinity difference, S– Increase flow, q

• But increasing S decreases q -> advective nonlinearity. Maximum qS at δ=0.5

25.02 Tk

HE S

Coupling effects

• Important that this model has different coupling of temperature and salinity to atmosphere. Alternatives:

• Temperature and salinity prescribed– Flow prescribed, single equilibrium

• Heat flux and salinity flux prescribed– Surface buoyancy flux prescribed, hence

equilibrium transport– Direction of flow determined by sign of surface

buoyancy flux

Summary

• 2-box model for thermohaline circulation with temperature imposed by atmosphere has two regimes:

• Above critical freshwater forcing, single thermally indirect solution, equatorward flow

• Below critical forcing, three solutions– Thermally indirect solution– Thermally direct solution, weak poleward flow– Thermally direct solution, strong poleward flow

• Tomorrow: look at stability


system:Stability of equilibrium solutions

Tim Henstock

Recap and plan

• Derived equilibrium states of simple 2-box model for thermohaline circulation, parameterised by δ and E

• Two regimes depending on critical freshwater forcing, with single or multiple equilibria and equatorward or poleward flow

• Today: investigate stability of the equilibria – what happens if we perturb forcing (HS) or salinity difference (S)?

Recap: equilibrium solutions

Equator-ward flow

Pole-ward flow

Weak flow

Strong flow

Multiple equilibria Single equilibrium

Recast equations (1)

• Previously:(18)hence(29)and(23)so(30)

0;1,02

kHTSST

Sq S

2

2

Tk

H

T

S

T

S S

0;1,02

kHTSST

Sq S

2

2

Tk

H

T

S

T

S S


• Put in dimensionless form:

(31)

and

(32)

• Sideways parabolas, opposite orientation. Intersect at δ=0 (no salinity difference) and δ =1 (no flow)

1;1 2E

1;1 2 E


• Put salinity conservation in dimensionless form:

(14)becomes(33)or(34)where use advective timescale 1/2kαT

SSTkHS S 22

T

S

T

S

Tk

H

T

S

dt

d

TkS

1

2

12

1E

Application to stability

• On equilibrium curve, trend in δ vanishes• To left of curve, E too small, δ decreases with

time• To right of curve, E too large, δ increases with

time• Note that for every δ there is a unique E, so

“left” and “right” unambiguous• Length of arrows based on rate of change of

δ

Stability of solutions

Implications

• Stability depends on branch we are on. • Move left from:

– Top branch (δ >1), trend back to stability– Bottom branch (δ <0.5), trend back to stability– Middle branch (0.5< δ <1), transition to bottom

branch

• Move right from:– Top branch (δ >1), trend back to stability– Bottom branch (δ <0.5), trend back to stability

(unless E becomes>0.25)– Middle branch (0.5< δ <1), transition to top branch

Interpretation

• Salinity-dominated state (δ>1, equatorward flow) always stable

• Strong flow thermally-dominated state (δ<0.5, poleward flow) always stable if it exists

• Weak flow thermally-dominated state (0.5<δ<1, poleward flow) unstable to small perturbations– Increase in forcing (E) leads to decrease in

response (δ)

Summary

• 2-box model for thermohaline circulation with temperature imposed by atmosphere has two regimes:


• Below critical forcing, three solutions– Thermally indirect solution, stable– Thermally direct solution, weak poleward flow,

unstable– Thermally direct solution, strong poleward flow,

stable


system:Feedbacks

Tim Henstock

Recap and plan

• Found three solutions to Stommel’s model– Thermally indirect solution, stable– Thermally direct solution, weak poleward

flow, unstable– Thermally direct solution, strong poleward

flow, stable

• Now investigate feedbacks, full solution

Linearised approximation (1)

• Return to original equations:

(9)

(13)

• Recast all as steady state value and perturbation

(37)

so

STkk

q

0

SqHSSS S 2212

qqqSSS ,


• From (9):

(38)

• NB that because T is external parameter

(39)

0;

0;

qSkqTkSSk

qSkqSSkTk

SSkTkqqq

Skq

Salinity conservation again

• Hence

(40)

since steady state value does not vary, and substituting steady state relation (15) we get

(41)

0:;0:;22

22

qqSSSkqH

SqHSSSS

S

S

SSSkSqS 22


• Now assume that close to equilibrium,(42)To get(43)• Coefficient of S’ negative:

– Perturbation exponentially damped, stable

• Coefficient of S’ positive:– Perturbation grows exponentially, unstable

qqSS ;

0:;0:;22 qqSSkSqS

Feedbacks (1)

• Each term in (43) represents a feedback• First term, mean flow feedback

– Works against an anomaly, always stabilising

• Second term, salinity transport feedback– Mean salinity difference x perturbation flow– Sign depends on steady-state flow

direction, destabilises thermally direct, but stabilises thermally indirect

Feedbacks (2)

• Thermally indirect/haline dominated mode stabilised by both feedbacks

• Thermally direct mode has positive and negative contributions; use flow law

(9)

to get

(44)

STkq

S

T

k

SSTkSSkSqS

212

2222

Feedbacks (3)

• Rewrite (44) for thermally direct mode in terms of stable dimensionless salinity

(44)• If δ<0.5, coefficient is negative, and

equilibrium is stable– Dominated by stabilising mean flow

• But if δ>0.5, coefficient is positive and equilibrium is unstable– Dominated by destabilising salinity transport

ST

kS

21

2

Full solution (1)

• Can solve (34) exactly in terms of E and dimensionless time (see handout)

• At large t, tend to one of the stable equilibria

• Can have long periods of little change followed by rapid transitions

Full solution (2)

• E=0.2

• Tend to stable states, can be rapid

Full solution (3)

• E=0.24

• Long meta-stability at δ=0.6

Full solution (4)

• E=0.26, ie only haline-dominated stable

• Long slow evolution, then rapid transition to haline-dominated

Summary (1)• 2-box model for thermohaline circulation with

temperature imposed by atmosphere has two regimes:


• Below critical forcing, three solutions– Thermally indirect solution, stable– Thermally direct solution, weak poleward flow,

unstable– Thermally direct solution, strong poleward flow,

stable

Summary (2)

• Time-dependent solution shows can have long periods of slow change, then rapid transition to different state

• Model predicts that transition from poleward to equatorward flow can result from changes in salinity

• Transition from equatorward to poleward flow not possible within this model framework

SOES6002 Module A, part 2: Multiple equilibria in the THC system: Governing equation Tim Henstock.

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Transcript of SOES6002 Module A, part 2: Multiple equilibria in the THC system: Governing equation Tim Henstock.