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 –T 1, T 2 temperature in each box –S 1, S 2 salinity in each box –q strength of circulation Assume that atmosphere controls: –T 1, T 2 –H S virtual salinity flux due to evaporation/ precipitation balance E, H S =S 0 E/D

Dynamics Flow law for q: (2) Equation of state (3) ie (5)

Conservation equations Temperature imposed externally, so no need to consider heat Salt: (6) (7) Note that box 1 imports S 2 and exports S 1, independent of the sign of q Also (6)+(7)=0, ie total salt is conserved

New approach Consider north-south (meridional) differences in the properties, and recast equations: (8) (NB deliberately different senses!) so (9)

Final equation Recast (6), (7) in terms of S (13) So, using (9) (14) Entire behaviour of model controlled by (14)

Summary 2-box model for thermohaline circulation Temperature imposed by atmosphere Simple dynamics, controlled by density differences Behaviour described by single equation

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

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:

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)

Case 1 (2): Hence (17) or (18) With solutions: (19)

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

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)

Case 2 (2): Hence (22) or (23) With solution: (24)

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 0, ie poleward flow, salinity driving force relative to external temperature driving force

Equilibrium solutions (2) Equator- ward flow Pole- ward flow

Equilibrium solutions (3): 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

Equilibrium solutions (4) Equator- ward flow Pole- ward flow Weak flow Strong flow

Equilibrium solutions (5): 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

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

SOES6002 Module A, part 2: Multiple equilibria in the THC 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 (H S ) 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)

Recast equations (2) Put in dimensionless form: (31) and (32) Sideways parabolas, opposite orientation. Intersect at δ=0 (no salinity difference) and δ =1 (no flow)

Recast equations (3) Put salinity conservation in dimensionless form: (14) becomes (33) or (34) where use advective timescale 1/2kαT

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 0.25) –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: Above critical freshwater forcing, single thermally indirect solution, equatorward flow Below critical forcing, three solutions –Thermally indirect solution, stable –Thermally direct solution, weak poleward flow, unstable –Thermally direct solution, strong poleward flow, stable

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

Linearised approximation (2) From (9): (38) NB that because T is external parameter (39)

Salinity conservation again Hence (40) since steady state value does not vary, and substituting steady state relation (15) we get (41)

Linearised approximation (3) 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

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)

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

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