Thermohaline circulation: a missing equation and its climate-change implications

Abstract

We formulate a box model of coupled ocean–atmosphere to examine the differential fields interactive with the thermohaline circulation (THC) and their response to global warming. We discern a robust convective bound on the atmospheric heat transport, which would divide the climate regime into warm and cold branches; but unlike the saline mode of previous box models, the cold state, if allowed, has the same-signed—though weaker—density contrast and THC as the present climate, which may explain its emergence from coupled general circulation models. We underscore the nondeterminacy of the THC due to random eddy shedding and apply the fluctuation theorem to constrain the shedding rate, thus closing the problem. The derivation reveals an ocean propelled toward the maximum entropy production (MEP) on millennial timescale (termed “MEP-adjustment”), the long timescale arising from the compounding effect of microscopic fluctuations in the shedding rate and their slight probability bias. Global warming may induce hysteresis between the two branches, like that seen in GCMs, but the cold transition is far more sensitive to the moistening than the heating effects as the latter would be countered by the hydrological feedback. The uni- or bi-modality of the current state—hence whether the THC may recover after the cold transition—depends on the global-mean convective flux and may not be easily assessed due to its observed uncertainty.

Appendices

Appendix A: Acronyms

EP:

Entropy production

FT:

Fluctuation theorem

GCM:

General circulation model

LW:

Longwave

MEP:

Maximum entropy production

MOC:

Meridional overturning circulation

NT:

Nonequilibrium thermodynamics

OLR:

Outgoing longwave radiation

SAT:

Surface-air temperature

SST:

Sea-surface temperature

SW:

Shortwave

THC:

Thermohaline circulation

Appendix B: Symbols

a:

Admittance

A:

Surface area spanned by cold ocean-box $( =1.9 \times {10^7}\; {\text{k}}{{\text{m}}^2})$

$B{o^{ - 1}}$ :

Inverse Bowen ratio over cold water ($=1$)

${C_d}$ :

Drag coefficient $(={10^{ - 3}})$

${C_{p,a}}$ :

Specific heat of surface air $(={10^3}\; {\text{JKg}}{ ^{ - 1}}{{\text{K}}^{ - 1}})$

${c_{p,o}}$ :

Specific heat of ocean $(=4.2 \times {10^{3 }}\;{\text{JKg}}{ ^{ - 1}}{{\text{K}}^{ - 1}})$

${F_w}$ :

Freshwater transport to the cold box

$g$ :

Gravitational acceleration ($=9.8\;{\text{m}}{{\text{s}}^{ - 2}}$)

K:

Mass exchange rate between ocean boxes (THC)

l:

Decorrelation distance of eddy shedding (=300 km)

L:

Latent heat of vaporization ($2.26 \times {10^6}\; {\text{J}}{\text{Kg}}^{ - 1}$) Length of subtropical front (=3000 km)

$q'$ :

Cold-box deficit of absorbed solar flux

${\bar q_c}$ :

Global-mean convective flux

${q_c}^\prime$ :

Cold-box deficit of convective flux

$\bar S$ :

Global-mean surface salinity (=35)

$\bar T$ :

Global-mean SST $( =15{ ^ \circ }{\text{C}})$

$T'$ :

Temperature deficit of cold ocean-box

${\bar T_a}$ :

Global-mean SAT

${T_a}^\prime$ :

Temperature deficit of cold atmosphere-box

${T_f}^\prime$ :

Freezing temperature

$|u'|$ :

Surface turbulent wind $( =7 {\text{m}}/{\text{s}})$

$\alpha$ :

Thermal expansion coefficient $(=1.7 \times {10^{ - 4}}{ ^ \circ }{{\text{C}}^{ - 1}})$

${\alpha ^*}$ :

Surface transfer coefficient $( \equiv {C_d}{\rho _a}{C_{p,a}}\left| {u'} \right|\left[ {1 + B{o^{ - 1}}} \right] = 14\; {\text{W}}{{\text{m}}^{ - 2}}\,\,{^ \circ}{{\text{C}}^{ - 1}})$

$\beta$ :

Saline contraction coefficient $(=7.6 \times {10^{ - 4}})$

$\rho '$ :

Density surplus of cold ocean-box.

${\rho _a}$ :

Surface air density ($=1\;{\text{Kg}}{{\text{m}}^{ - 3}}$)

${\rho _o}$ :

Ocean density ($={10^3}\; {\text{Kg}} {{\text{m}}^{ - 3}}$)

$\sigma$ :

Ocean entropy production rate.

${\sigma _a}$ :

Atmosphere entropy production rate

$\mu$ :

Moisture-content parameter (=0.3)

Appendix C: Scale definitions

$$[q'] = 100 \,{\text{W}} {{\text{m}}^{ - 2}}$$

$$\left[ {T'} \right] \equiv [q']/{\alpha ^*} = {7.1^ \circ }{\text{C}}$$

$$\left[ {S'} \right] = \alpha \left[ {T'} \right]/\beta = 1.6$$

$$\left[ {\rho '} \right] = {\rho _o}\alpha [T'] = 1.36\; {\text{Kg}}{{\text{m}}^{ - 3}}$$

$$[K] = {\left( {2{\rho _o}{c_{p,o}}[T']} \right)^{ - 1}}\left[ {q'} \right]A = 31\; Sv$$

$$\left[ a \right] = \left[ K \right]/\left[ {\rho '} \right]$$

$$[{F_w}] = 2\left[ K \right]\left[ {S'} \right]/\bar S = 2.8\; Sv$$

$$\left[ \sigma \right] = 2{\bar T^{ - 2}}\left[ {q'} \right][T']$$

$$\left[ {{\sigma _a}} \right] = 2{\bar T_a}^{ - 2}\left[ {q'} \right][T']$$

Appendix D: Moisture transport

By considering the hydrological cycle, Ou (2007) has linked the moisture transport at mid-latitudes to the atmospheric heat transport $F_a^*$ (starred for dimensional variables). Adjusting this moisture transport for the freshwater input $F_w^*$ to the cold box because of the differing catchment (${A_c}$) and ocean (A) areas, we have

$$F_w^* = {\mu ^*}F_a^*{A_c}/A,$$

(28)

where

$${\mu ^*} = \frac{1}{{{\rho _o}L(1 + 1/Bo_m^{ - 1})}}.$$

(29)

In the above, $Bo_m^{ - 1}$ (the inverse “meridional” Bowen ratio) reflects the moisture content of the air column and is of the form

$$Bo_m^{ - 1} = const. \times e/T,$$

(30)

where T is the mean SAT in the frontal zone (approximated by the global-mean SAT) and e, its saturation vapor pressure, a relation that has been validated by observation (Ou 2007). When nondimensionalized, (28) becomes

$${F_w} = \mu {q_c}^\prime ,$$

(31)

where

$$\mu = \frac{c}{{1 + 1/b}}$$

(32)

with

$$b = Bo_m^{ - 1}$$

(33)

and

$$c = \frac{{{A_c}}}{A} \cdot \frac{{{C_{p,o}}\beta \overline{\overline S} }}{{L\alpha }}.$$

(34)

For the standard case, we use $b \approx 0.5$ (Ou 2007, his Fig. 4), and setting${A_c}/A \approx 3$, we obtain $c \approx$ 0.87, so $\mu \approx$ 0.3. As seen in (25), $\mu$ yields the inverse density-ratio of the current climate, which is observed to be about 0.3 (Tippins and Tomczak 2003), so our selection of ${A_c}/A$, given its high uncertainty, is partly to match this observation.

Appendix E: Response to freshwater perturbation

Subjected to the external freshwater perturbation w′, the salinity balance (9)–(10) states

$$\mu {q_c}^\prime + w' = KS'.$$

(35)

Eliminating T′, S′ and $\rho '$ from (1), (6)–(14), we derive, for the warm branch,

$$w' = \frac{{2K - \mu }}{{2K + 1}} - \frac{{{K^2}}}{a},$$

(36)

where the admittance a can be seen from (13), (14) and (25) to be given by $a = {[2\left( {1 - \mu } \right)]^{ - 1}}$ and, for the cold branch,

$$w' = 1 - \left( {1 + \mu } \right){\bar q_c} - \frac{{{K^2}}}{a}.$$

(37)

These equations allow us to calculate, in a reverse manner, the perturbation $w'$ given K, as plotted in Fig. 3.

Appendix F: Moistening parameter

With global warming, the inverse “meridional” Bowen ratio b of (33) and (30) would be perturbed as

$$\begin{gathered} {\text{ln'}}b = {\text{ln'}}e - {\text{ln'}}T \hfill \\ = \left( {\frac{L}{{RT}} - 1} \right)\operatorname{l} n'T\;\quad \left( {{\text{Clausius}} - {\text{Clapeyron equation}}} \right) \hfill \\ = {b_1}T', \hfill \\ \end{gathered}$$

(38)

where

$${b_1} \equiv \left( {\frac{L}{{RT}} - 1} \right) \cdot \frac{1}{T}$$

(39)

and the temperature can be approximated by the global-mean SAT (see the discussion following [10]). Setting the latter to ${288^ \circ }K$ yields ${b_1} \approx$ 0.092. From (32), we derive the perturbation in$\mu$,

$$\begin{aligned} {\text{ln'}}\mu & = \frac{1}{{1 + b}}{\text{ln'}}b \\ & \approx \frac{{{b_1}}}{{1 + {b_0}}}T' \\ \end{aligned}$$

(40)

where ${b_0}$ is the unperturbed value of b. Keeping only the first-order terms in the perturbation and replacing $T'$ with the symbol $\delta {\bar T_a}$, we arrive at

$$\mu \approx {\mu _0} + {\mu _1}\delta {\bar T_a}$$

(41)

where ${\mu _0} = 0.3$ (“Appendix D”) and

$${\mu _1} = \frac{{{\mu _0}{b_1}}}{{1 + {b_0}}}.$$

(42)

For the standard case, ${\mu _1} \approx 0.02$.

Appendix G: Heating parameter

The cold-box deficit of the absorbed solar flux from its global-mean is

$${q^*} = {\bar q_s}^*\left( {1 - {{\bar a}_l}} \right) - q_s^*(1 - {a_l}),$$

(43)

where stars indicate dimensional variables, $q_s^*$ is the incident “solar” flux for the cold box and ${a_l}$is its surface albedo, both over-barred for their global-mean counterparts. As their representative values, we set ${\bar q_s}^* = 300\; {\text{W}} {{\text{m}}^{ - 2}}$, $q_s^* = 200\; {\text{W}} {{\text{m}}^{ - 2}},$ $\bar{a}_{l} = .3$ and ${a_l} = 0.45$ (Robock 1980, his Fig. 20) to yield${q^*} = 100\; {\text{W}} {{\text{m}}^{ - 2}}$, which sets the forcing scale [q’]. A global warming of $~\delta \bar{T}_{a}$ (in $^ \circ {\text{C}}$) would reduce the cold-box albedo by $\delta {a_l}$ hence the global-mean albedo by $\delta \bar{a}_{l} = \delta a_{l}$/2 since the cold box spans half the global surface. Dividing (43) by [q′], the non-dimensionalized forcing is then

$$q^{\prime} = {q_0}^\prime - {q_1}^\prime \delta {\bar T_a},$$

(44)

where subscripts 0 and 1 denote its unperturbed value and its perturbation per degree-warming with ${q_0}^\prime = 1$ and

$${q_1}^\prime = \frac{1}{{\left[ {q'} \right]}}\left( {q_s^* - \frac{1}{2}{{\bar q}_s}^*} \right)\frac{{\delta {a_l}}}{{\delta {{\bar T}_a}}}.$$

(45)

Based on Robock (1980, his Table 11 for B in high northern latitudes), the fractional ice cover decreases by about 0.03 per degree-warming, and taking the albedo difference between the ice-covered and ice-free surface to be 0.7, we estimate $\delta a_{l} /~~\delta \bar{T}_{a} \approx 0.03 \times 0.7 = 0.021$. Applying foregoing radiative fluxes to (45) yields ${q_1}^\prime \approx 0.01$.

Appendix H: Response to global warming

Eliminating T′, S′ and $\rho '$ from (1), (6)–(14) yields, for the warm branch,

$$q'\left( {1 - \frac{\mu }{{2K}}} \right) = \frac{K}{a}\left( {\frac{1}{2} + K} \right).$$

(46)

Applying (41), (44) and retaining only linear terms in $\delta {\bar T_a}$ (the higher-order terms are more than an order-of-magnitude smaller) yields

$$\delta {\bar T_a} \approx C/B,$$

(47)

where

$$C = {q_0}^\prime \left( {1 - \frac{{{\mu _0}}}{{2K}}} \right) - \frac{K}{a}\left( {\frac{1}{2} + K} \right),$$

(48)

$$B = {q_1}^\prime \left( {1 - \frac{{{\mu _0}}}{{2K}}} \right) + \frac{{{q_0}^\prime {\mu _1}}}{{2K}},$$

(49)

which allows the calculation of $\delta {\bar T_a}$ given K. The counterpart to (46) for the cold branch is

$$q' - \left( {1 + \mu } \right){\bar q_c} = {K^2}/a.$$

(50)

We have again (47) but with

$$C = {q_0}^\prime - \left( {1 + {\mu _0}} \right){\bar q_c} - {K^2}/a,$$

(51)

$$B = {q_1}^\prime + {\mu _1}{\bar q_c}.$$

(52)