The Stommel two-box model [37] is one of the first mathematical formalism employed to study the dynamical characteristics of AMOC to nonlinear climate forcings. This section references the work in [26] and sketches its main derivations and analytical results relative to the tipping point behaviour.

We divide the Atlantic into two vertically mixed boxes: the high latitudes part of the Atlantic Ocean, the red box, and the low latitudes one. They are connected by AMOC-driven volume transport $q$. The circulation is governed by the density difference:

$q=k\left({\rho }_1-{\rho }_2\right)=k\left(-{\alpha }_t\text{Δ}T+{\beta }_s\text{Δ}S\right),$ (1)

where $\text{Δ}T=T_1-T_2<0$ (fixed temperature gradient), $\text{Δ}S=S_1-S_2$ (salinity difference), and $k$ is a proportionality constant.

Assuming equal box volumes $v_1=v_2=v$, the salt conservation equations under freshwater forcing $F_s$ become:

$v\frac{\text{d}{S}_1}{\text{d}t}=\left|q\right|\left(S_2-S_1\right)-F_s,$ (2)

$v\frac{\text{d}{S}_2}{\text{d}t}=\left|q\right|\left(S_1-S_2\right)+F_s.$ (3)

Subtracting these yields the governing equation for salinity difference:

$-v\frac{\text{dΔ}S}{\text{d}t}=2\left|q\right|\text{Δ}S+2F_s.$ (4)

Substituting $q$ from Equation (4) and defining rescaled variables $X={\alpha }_T\text{Δ}T$ and $Y={\beta }_R\text{Δ}S$, we have:

$\left|Y-X\right|Y=-\frac{{\beta }_{s}F}{k}.$ (5)

For $Y<X$, the equation becomes:

$Y^2-XY-\frac{{\beta }_s{F}_s}{k}=0,$

with one valid solution:

$Y=\frac{X}{2}-\frac{1}{2}{\left(X^2+4\frac{{\beta }_s{F}_s}{k}\right)}^{1/2}.$

For $Y>X$, the equation becomes:

$Y^2-XY+\frac{{\beta }_s{F}_s}{k}=0,$

with solutions:

$Y=\frac{X}{2}\pm \frac{1}{2}{\left(X^2-4\frac{{\beta }_s{F}_s}{k}\right)}^{1/2}.$

Note that the plus solution for Y in the first case is positive and is therefore not consistent with the assumption $Y<X$ used to obtain that solution, hence the system exhibits up to three equilibria depending on $F_s$.

Equation (5) has up to three solutions (see Figure 3 ):

• Strong AMOC (green branch): High $q$ state maintained by salt-advection feedback.

• Unstable AMOC (blue branch): Intermediate state prone to disturbances.

• Collapsed AMOC (red branch): Weak reversed flow under strong freshwater forcing.

Analysis shows that the blue branch solution is unstable: perturbation grows exponentially in this case, whereas deviations from a stable state decrease. It leads to jumps when crossing critical forcing thresholds because of the structure of the bifurcation.

Gradual freshwater forcing increase (e.g., from ice melt under warming) weakens AMOC along the green branch until collapse occurs at $F\approx 1.2$ m/s. Importantly, reducing F to less than the threshold value of about $F\approx 1.2$ m/s is not sufficient to recover the initial state; a lower $F_s$ needs to be sustained in order to obtain this state. The non-reversible nature of this transition results from the quadratic nonlinearity found in Equation (5), typical of abrupt tipping mechanisms in the climate system.

We divide the world ocean into three boxes: the blue box represents the high latitudes part of the Atlantic Ocean, the red box, the low latitudes one and the black box, the global ocean except the Atlantic Ocean. We let the ocean box temperature $T_1,T_2,T_3$ to be constants. Assuming the volumes of the ocean boxes are constant, let $v_1,v_2,v_3$ represent the volume of the blue box, the red box and the black box.

Denote the circulation between the red box and the blue box as $q_1$, the circulation between the red box and the black box as $q_2$. The unknowns in the three-box model is the three boxes and the circulation $q_1,q_2$. See Figure 4 .

Similar to that in the two-box model, denote the freshing effect caused by glacier melting and excess precipitation in the blue box and the black box as $-F_{s1}$ and $-F_{s3}$. Considering both the salt transfer between the red box and the blue box, the red box and the black box, we derived

$v_1\frac{\text{d}{S}_1\left(t\right)}{\text{d}t}=\left|q_1\right|\left(S_2-S_1\right)-F_{s1}$

$v_2\frac{\text{d}{S}_2\left(t\right)}{\text{d}t}=\left|q_1\right|\left(S_1-S_2\right)+\left|q_2\right|\left(S_3-S_2\right)+F_{s2}$

$v_3\frac{\text{d}{S}_3\left(t\right)}{\text{d}t}=\left|q_2\right|\left(S_2-S_3\right)-F_{s3}$

Let

$\text{Δ}{T}_1=T_1-T_2<0,\text{Δ}{T}_2=T_3-T_2<0$

$\text{Δ}{S}_1=S_1-S_2,\text{Δ}{S}_2=S_3-S_2$

The circulation is assumed to be proportional to the density difference between the boxes, so $q_i,i=1,2$ can be written as $k_i\left(-{\alpha }_t\text{Δ}{T}_i+{\beta }_s\text{Δ}{S}_i\right),i=1,2$. Therefore, we derived

$v_1\frac{\text{d}{S}_1\left(t\right)}{\text{d}t}=\left|k_1\left(-{\alpha }_t\left(T_1-T_2\right)+{\beta }_s\left(S_1-S_2\right)\right)\right|\left(S_2-S_1\right)-F_{s1}$ (6)

$\begin{array}{c}{v}_2\frac{\text{d}{S}_2\left(t\right)}{\text{d}t}=\left|k_1\left(-{\alpha }_t\left(T_1-T_2\right)+{\beta }_s\left(S_1-S_2\right)\right)\right|\left(S_1-S_2\right)\\ +\left|k_2\left(-{\alpha }_t\left(T_3-T_2\right)+{\beta }_s\left(S_3-S_2\right)\right)\right|\left(S_3-S_2\right)+F_{s2}\end{array}$ (7)

$v_3\frac{\text{d}{S}_3\left(t\right)}{\text{d}t}=\left|k_2\left(-{\alpha }_t\left(T_3-T_2\right)+{\beta }_s\left(S_3-S_2\right)\right)\right|\left(S_2-S_3\right)-F_{s3}$ (8)

From (6) - (8), assuming that

$F_{s2}=F_{s1}+F_{s3}$

In this section, we take the division to be $v_1=v_2$. According to [38] , the volume of the global ocean is approximately 4 times the volume of Atlantic. Thus:

$v_1=v_2=\frac{v_3}{6}$

Denote

$y_1={\beta }_s\text{Δ}{S}_1,y_2={\beta }_s\text{Δ}{S}_2,x_1={\alpha }_t\text{Δ}{T}_1,x_2={\alpha }_t\text{Δ}{T}_2,$

We have

$\frac{\text{d}{y}_1}{\text{d}t}=-\frac{2k}{v_1}\left|y_1-x_1\right|y_1-\frac{k}{v_1}\left|y_2-x_2\right|y_2-\frac{\left(F_{s1}+F_{s2}\right){\beta }_s}{v_1}$ (9)

$\frac{\text{d}{y}_2}{\text{d}t}=-\frac{7k}{6v_1}\left|y_2-x_2\right|y_2-\frac{k}{v_1}\left|y_1-x_1\right|y_1-\left(\frac{F_{s3}}{6v_1}+\frac{F_{s2}}{v_1}\right){\beta }_s$ (10)

In a steady state, we have

$\frac{\text{d}{y}_1}{\text{d}t}=0,$

$\frac{\text{d}{y}_2}{\text{d}t}=0,$

Noticing that $F_{s2}=F_{s1}+F_{s3}$,

From (9) − 2 * (10), we have

$0=\frac{\text{d}{y}_1}{\text{d}t}-2\frac{\text{d}{y}_2}{\text{d}t}=\frac{4}{3}\left(\frac{k}{v_1}\left|y_2-x_2\right|y_2+\frac{{\beta }_s\cdot F_{s3}}{v_1}\right)$ (11)

Similarly,

$\frac{k}{v_1}\left|y_1-x_1\right|y_1+\frac{F_{s1}\cdot {\beta }_s}{v_1}=0$ (12)

Thus, we have separated the entire large system into two independent subsystems. Recall that in the two-box model, there are three solutions when $x^2>4\frac{{\beta }_s{F}_s}{k}$ and one solution when $x^2<4\frac{{\beta }_s{F}_s}{k}$. Similar to this approach, we categorize the problem into the following four cases.

Case 1: $x_1^2>4\frac{{\beta }_s{F}_{s1}}{k}$ and $x_2^2>4\frac{{\beta }_s{F}_{s2}}{k}$

Similar to the two-box case, denote:

$y_i^{\left(1\right)}=\frac{x}{2}-\frac{1}{2}\sqrt{x^2+4\frac{{\beta }_s{F}_{si}}{k}},$

$y_i^{\left(2\right)}=\frac{x}{2}-\frac{1}{2}\sqrt{x^2-4\frac{{\beta }_s{F}_{si}}{k}},$

$y_i^{\left(3\right)}=\frac{x}{2}+\frac{1}{2}\sqrt{x^2-4\frac{{\beta }_s{F}_{si}}{k}},$

for $i=1,2$.

(We will also use this notation in Case 2, Case 3, and Case 4.)

It can be verified that $y_i^{\left(j\right)},j=1,2,3$ are the solutions for the function $\frac{k}{v_i}\left|y_i-x_i\right|y_i+\frac{F_{si}\cdot {\beta }_s}{v_i}=0,i=1,2$.

As a result, there are up to $3\times 3=9$ steady states, i.e., combinations of $\left(y_1^{\left(j_1\right)},y_2^{\left(j_2\right)}\right)$ for $j_1,j_2=1,2,3$.

The Jacobian analysis shows that 4 of these steady states are stable:

$\left(y_1,y_2\right)=\left(y_1^{\left(1\right)},y_2^{\left(1\right)}\right),\left(y_1^{\left(1\right)},y_2^{\left(3\right)}\right),\left(y_1^{\left(3\right)},y_2^{\left(1\right)}\right),\left(y_1^{\left(3\right)},y_2^{\left(3\right)}\right).$

And they represent the four main modes of the circulation, see Figure 5 .

Case 2: $x_1^2<4\frac{{\beta }_s{F}_{s1}}{k}$ and $x_2^2>4\frac{{\beta }_s{F}_{s2}}{k}$

Similarly, it can be inferred that there are 3 steady states in total:

$\left(y_1,y_2\right)=\left(y_1^{\left(1\right)},y_2^{\left(j\right)}\right),j=1,2,3,$

with 2 of them being stable:

$\left(y_1,y_2\right)=\left(y_1^{\left(1\right)},y_2^{\left(1\right)}\right),\left(y_1^{\left(1\right)},y_2^{\left(3\right)}\right).$

Case 3: $x_1^2>4\frac{{\beta }_s{F}_{s1}}{k}$ and $x_2^2<4\frac{{\beta }_s{F}_{s2}}{k}$

Similarly, it can be inferred that there are 3 steady states in total:

$\left(y_1,y_2\right)=\left(y_1^{\left(j\right)},y_2^{\left(1\right)}\right),j=1,2,3,$

with 2 of them being stable:

$\left(y_1,y_2\right)=\left(y_1^{\left(1\right)},y_2^{\left(1\right)}\right),\left(y_1^{\left(3\right)},y_2^{\left(1\right)}\right).$

Case 4: $x_1^2<4\frac{{\beta }_s{F}_{s1}}{k}$ and $x_2^2<4\frac{{\beta }_s{F}_{s2}}{k}$

Similarly, it can be inferred that there exists only one steady state:

$\left(y_1,y_2\right)=\left(y_1^{\left(1\right)},y_2^{\left(1\right)}\right)$

And the steady state is stable.

Up to this point, we have discussed the steady-state solutions of this ordinary differential equation. However, for a given initial value, will the solution trajectory ultimately converge to one of these steady-state points? The following proof guarantees this.

Theorem 1 (Stability Analysis) There exists $a_1,a_2\in R$ such that

$\underset{t\to \infty }{\text{lim}}\left|y_1-a_1\right|=0$ and $\underset{t\to \infty }{\text{lim}}\left|y_2-a_2\right|=0$.

Proof 3.1 We only prove the case $x_1^2\ge 4\frac{F_{s1}\cdot {\beta }_s}{V},x_2^2\ge 4\frac{F_{s2}\cdot {\beta }_s}{V}$:

Denote

$\begin{array}{l}f\left(y\right)=-\frac{k}{v}\left|\left(y-x_1\right)\right|y-\frac{F_{s1}\cdot {\beta }_s}{v},\\ g\left(y\right)=-\frac{k}{v}\left|\left(y-x_2\right)\right|y-\frac{F_{s2}\cdot {\beta }_s}{v},\end{array}$

Thus

$\begin{array}{l}\frac{\text{d}{y}_1}{\text{d}t}=2f\left(y_1\right)+g\left(y_2\right),\\ \frac{\text{d}{y}_2}{\text{d}t}=f\left(y_1\right)+\frac{7}{6}g\left(y_2\right).\end{array}$

First consider the system $\frac{\text{d}y}{\text{d}t}=2f\left(y\right)$.

Similar to the two-box model, because $x_1^2\ge 4\frac{F_{s1}\cdot {\beta }_s}{v}$, the system has three steady states. Let $b_1<b_2$ be the stable point of the system:

$\frac{\text{d}y}{\text{d}t}=2f\left(y\right).$

$b_3$ is the unstable steady point. (Review this in the two-box case.)

Let $c_1<c_2$ be the stable point of the system:

$\frac{\text{d}y}{\text{d}t}=2g\left(y\right).$

$c_3$ is the unstable steady point. (Review this in the two-box case.)

Using these notations, noticing that in the three box model, when $y_1\le b_1$, $f\left(y_1\right)\ge 0$; $b_1\le y_1\le b_3$, $f\left(y_1\right)\le 0$, so when $y_1\le b_3$:

$\frac{\text{d}}{\text{d}t}\left|y_1-b_1\right|=-2\left|f\left(y_1\right)\right|+g\left(y_2\right)\mathrm{sgn}\left(y_1-b_1\right)\le -2\left|f\left(y_1\right)\right|+\left|g\left(y_2\right)\right|$ (13)

Similarly:

-If $y_1\ge b_3$, then $\frac{\text{d}}{\text{d}t}\left|y_1-b_2\right|\le -2\left|f\left(y_1\right)\right|+\left|g\left(y_2\right)\right|$ (14)

-If $y_2\le c_3$, then $\frac{\text{d}}{\text{d}t}\left|y_2-c_1\right|\le -2\left|f\left(y_1\right)\right|+\left|g\left(y_2\right)\right|$ (15)

-If $y_2\ge c_3$, then $\frac{\text{d}}{\text{d}t}\left|y_2-c_2\right|\le -2\left|f\left(y_1\right)\right|+\left|g\left(y_2\right)\right|$ (16)

Let

$h\left(t\right)=\text{min}\left\{\left|y_1-b_1\right|+b_3-2b_2+b_1,\left|y_1-b_2\right|\right\}+\text{min}\left\{\left|y_2-c_1\right|+c_3-2c_2+c_1,\left|y_2-c_2\right|\right\}$

(We take the form $\text{min}\left\{\left|y_1-b_1\right|+b_3-2b_2+b_1,\left|y_1-b_2\right|\right\}$ to ensure the function strictly decreases.)

So, $h$ is continuous, and

from (13) and (14):

$\frac{\text{d}}{\text{d}t}\text{min}\left\{\left|y_1-b_1\right|+b_3-2b_2+b_1,\left|y_1-b_2\right|\right\}\le -2\left|f\left(y_1\right)\right|+\left|g\left(y_2\right)\right|$

from (15) and (16):

$\frac{\text{d}}{\text{d}t}\text{min}\left\{\left|y_2-c_1\right|+c_3-2c_2+c_1,\left|y_2-c_2\right|\right\}\le -\frac{7}{6}\left|g\left(y_2\right)\right|+\left|f\left(y_1\right)\right|$

Thus:

$\begin{array}{c}\frac{\text{d}h\left(t\right)}{\text{d}t}\le -\frac{1}{6}\left|g\left(y_2\right)\right|-\left|f\left(y_1\right)\right|\\ \le -\frac{1}{13}\left(\left|2f\left(y_1\right)+g\left(y_2\right)\right|+\left|f\left(y_1\right)+\frac{7}{6}g\left(y_2\right)\right|\right)\\ =-\frac{1}{13}\left(\left|\frac{\text{d}{y}_1}{\text{d}t}\right|+\left|\frac{\text{d}{y}_2}{\text{d}t}\right|\right)\end{array}$

Since $\frac{\text{d}h\left(t\right)}{\text{d}t}\le 0$, and noticing that:

$h\left(t\right)\ge -\left|b_3-2b_2+b_1|-|c_3-2c_2+c_1\right|,$

we have:

${\displaystyle {\int }_0^{\infty }\left|\frac{\text{d}{y}_1}{\text{d}t}\right|\text{d}t}+{\displaystyle {\int }_0^{\infty }\left|\frac{\text{d}{y}_2}{\text{d}t}\right|\text{d}t}\le 13\left(\left|h\left(0\right)\right|+\left|b_3-2b_2+b_1\right|+\left|c_3-2c_2+c_1\right|\right)$

Therefore, there exists $a_1,a_2$ such that:

$\underset{t\to \infty }{\text{lim}}\left|y_1-a_1\right|=0$ and $\underset{t\to \infty }{\text{lim}}\left|y_2-a_2\right|=0.$

From the previous section, we can see that for a steady state, the number of distinct value q1 may take is completely determined by $F_{s1}$.

We aim to study the solutions of $\text{Δ}{S}_1$ and $\text{Δ}{S}_2$ when given $F_{s1}$, $F_{s3}$, and initial values of $\text{Δ}{S}_1$, $\text{Δ}{S}_2$, as well as which steady-state solutions they will ultimately converge to. Here, we find that the influence of $F_{s3}$ on the entire system is non-negligible. Specifically, for given initial values and $F_{s1}$, even when only $F_{s3}$ is varied, the final state of $\text{Δ}{S}_1$ may exhibit fundamentally different characteristics.

For instance, when setting $F_{s1}=1\text{m}/\text{y}$ with $F_{s3}=0.18\text{m}/\text{y}$, compared to $F_{s1}=1\text{m}/\text{y}$ and $F_{s3}=0.2\text{m}/\text{y}$, we notice great dissimilarity of the $F_{s1}$ temporal evolution curves, see Figure 6 .

We notice that though when we take $F_{s1}=1,F_{s3}=0.2$, AMOC collapse, contrary to that of $F_{s1}=1,F_{s3}=0.18$.

Next, we show the minimum $F_{s1}$ value required to induce AMOC collapse under the initial conditions of $\text{Δ}{S}_3=-1$, $\text{Δ}{S}_1$ at its first steady state, and $F_{s3}$ increasing from 0 to 0.3, see Figure 7 .

We note that when $F_{s3}<0.186$, the critical value of $F_{s1}$ that causes circulation 1 to collapse is 1.186. This occurs precisely because when $F_{s1}>1.186$, in a steady state, circulation 1 has only one steady-state solution.

When we slowly increase the number of $F_{s3}$, passing by 1.86, the corresponding $F_{s1}$ decreases sharply. Similar to the idea in [26] , when we gradually increase $F_{s3}$, the overall number of solutions switches from 3 to 1, so $\text{Δ}{S}_2$ should switch to that solution.

Recalling the equation $\frac{\text{d}{y}_1}{\text{d}t}=-\frac{2k}{v_1}\left|y_1-x_1\right|y_1-\frac{k}{v_2}\left|y_2-x_2\right|y_2-\frac{\left(F_{s1}+F_{s2}\right){\beta }_s}{v_1}$. When $\text{Δ}{S}_2$ gradually decreases from its initial value to that solution, $\left(\frac{k}{v_1}\left|y_2-x_2\right|y_2+\frac{{\beta }_s\cdot F_{s3}}{v_1}\right)<0$. Heuristically speaking, this negative term becomes the dominant part on right hand side, making $\frac{\text{d}{y}_1}{\text{d}t}<0$, thereby driving $\text{Δ}{S}_1$ to decrease towards its second steady state point (see Figure 8 for the steady states of $q_1$ and $q_2$ corresponding to $F_{s1}$ and $F_{s3}$).

When $F_{s3}>1.86$, the relationship is subtle. From observation, the graph of the function is almost a straight line. The least squares method was applied to approximate this plot (see Figure 9 ), from which we derived the expression

$F_{s1}=1.3655-1.9846F_{s3}.$

As we have seen that the introduction of a third box enables the AMOC to “bypast” one of the robust steady states present in the two-box model. The physical extrapolation of this result reveals that the Pacific becomes a dynamic booster, imparting sufficient inertia to the system so that it cannot settle into the original potential wells.

In the traditional two-box model, transitioning from one steady state to another requires overcoming an energy barrier, acting as a kind of “sticky” hysteresis. However, in the three-box model, the inclusion of the Pacific introduces a huge volume and salinity capacity, endowing the system with a “kinetic flywheel” effect. When the system begins to evolve from one state to another under external forcing, the established salinity gradient between the Atlantic and Pacific drives sustained circulation and exchange. The momentum of this process prevents the system from pausing at the intermediate equilibrium points that existed before—much like a swing being pushed continuously does not come to rest at the midpoint but swings directly from one high point to another, thereby “bypass” the seemingly robust intermediate steady states that would have existed without the Pacific.

This phenomenon can be analogized to the formation process of “explosive cyclogenesis” in weather systems. During cyclogenesis, when certain conditions reach a critical threshold, the system does not slowly approach equilibrium. Instead, it gains immense inertia, rapidly skipping over weak stable phases and entering a period of rapid intensification or “explosion”. Similarly, in our three-box model, the additional capacity provided by the third box provides the AMOC with similar inertia during its state evolution, causing it to bypass its original steady-state structure.

As is well known from modelling, the Atlantic Meridional Overturning Circulation (AMOC) has more than one steady-state mechanism available. During D-O events, the vigorous AMOC brings increased quantities of warm water mass into the North Atlantic Ocean and Arctic Ocean, increasing the freshwater transport from Arctic to Northern Hemisphere landmasses as well as the subsequent, amplified freshwater forcing [39] [40] . As discussed in the Stommel’s Model in Section 2, when the amplitude of the freshwater forcing passes some critical threshold, the stable, strong, steady state vanishes, and the AMOC switches to the weak steady state. In this weak state, the northward heat transport is substantially reduced relative to the strong state, leading to a decrease in freshwater runoff and hence freshwater forcing. If the forcing becomes weak enough for the upstream process to cease and fresh water no longer moves northward, the strong AMOC state can be reinitiated, bringing about a complete cyclical transition through this means between the two distinct states.

In recent research [36] , the authors reconstructed a two-box AMOC model by introducing a temporal delay in the oceanic transport between the boxes through a delay function, demonstrating the self-sustained multidecadal variability of AMOC. We hypothesize that the mechanism of Dansgaard-Oeschger (D-O) events may share similar principles. Notably, the polar see-saw hypothesis discussed in [12] [14] suggests that during deepwater formation in polar regions, substantial changes (e.g., during intense glacial growth or depletion) require prolonged periods to manifest their effects in the opposite hemisphere.

This implies that when a strong AMOC resumes, Arctic runoff reduction shows delayed response. Consequently, the strong AMOC persists while continually “charging” the Arctic ice system with energy. When the AMOC’s influence eventually reaches the Arctic, the resultant dramatic increase in runoff forces the AMOC to collapse into a weak state. Due to this accumulated energy reserve, the elevated Arctic runoff persists even after AMOC weakening, maintaining the weak AMOC phase until the stored energy is fully depleted.

We suppose the volume of the volume, salinity and temperature for the Arctic are $V_1,S_1,T_1$, respectively, and those of the high latitude Atlantic and low latitude Atlantic are $V_2,S_2,T_2$ and $V_3,S_3,T_3$. Given the extended duration of Dansgaard-Oeschger (D-O) events, we explicitly consider both temperature and salinity as dynamic variables in our model framework. See Figure 10 for our model.

Our analysis begins with the freshwater forcing. The freshwater forcing primarily consists of two components: runoff from ice sheets and net rainfall. We assume the fresh water force for the Arctic and low-latitude Atlantic is dominated by the runoff of the ice-sheet and net rainfall, respectively, while the freshwater force for the high-latitude is dominated by both runoff from ice-sheet and net rainfall.

Firstly, let’s assess the effect of ice-sheet runoff on marine systems. Let R represent the runoff rate, and given freshwater’s near-zero salinity and temperature, we establish the core salinity and heat balance equation for a simple box:

$v\frac{\text{d}S}{\text{d}t}=-SR,v\frac{\text{d}T}{\text{d}t}=-TR,$

In reality, since the runoff R is much smaller than the ocean volumes, we assume that ocean volumes are constants. The volume of the Arctic Ocean is approximately 1.9 × 10⁷ km³. The volume of the North Atlantic Ocean is approximately 1.6 × 10⁸ km³ [38] . In this section, we define the partition as $v_1=4\times {10}^7{\text{km}}^3$ and $v_2=1.2\times {10}^8{\text{km}}^3$.

Besides, in D-O event, fresh water runs to the high latitude North Atlantic and the Arctic [41] , denote them by $R_N,R_A$, respectively. We take the ratio to be 2:1:

$R_A=\frac{R}{3},R_N=\frac{2}{3}R$

Again, let’s consider the effect of net rainfall. For simplicity, we assume the effect of net rainfall on high-latitude North Atlantic and low-latitude North Atlantic is constants. Denote it by $-Fs$ and $Fs$, respectively.

Let’s move on to the effect of AMOC on the ice-sheet runoff R. Varying meltwater runoff to the North Atlantic can occur in association with fluctuations of the ice margin [33] . There are a number of inter-related considerations here. We will use the result in [33] that the runoff R will depend linearly on AMOC. Besides, our framework incorporates a temporal lag (denoted as $d_1$) between the low-latitude Atlantic meridional overturning circulation and its impact on ice-sheet runoff dynamics:

$R=R_0+a\cdot q_2\left(t-d_1\right)$

Estimates of the period of temporal lag vary and one typical estimate is 400 years and we will illustrate it in the next section. Regarding the selection of parameters $R_0,a$ and $Q_0$, we refer to the work in [35] . We will use $R_0~0.1\text{Sv}$, $a~0.01$ and $Q_0~20\text{Sv}$.

For the volume transport $q_1,q_2$, we assume that they are proportional to the density difference between the boxes. Additionally, we incorporate another temporal lag (denoted as $d_2$) to characterize the mean advective time delay, a choice for $d_2$ is about 20 years:

$q_1=k_1\left(-{\alpha }_t\left(T_1\left(t-d_2\right)-T_2\left(t-d_2\right)\right)+{\beta }_s\left(S_1\left(t-d_2\right)-S_2\left(t-d_2\right)\right)\right)$

$q_2=k_2\left(-{\alpha }_t\left(T_2\left(t-d_2\right)-T_3\left(t-d_2\right)\right)+{\beta }_s\left(S_2\left(t-d_2\right)-S_3\left(t-d_2\right)\right)\right)$

The dynamic equations of salinity for our three-box model become:

$v_1\frac{\text{d}{S}_1\left(t\right)}{\text{d}t}=\left|q_1\right|\cdot \left(S_2\left(t-d_2\right)-S_1\left(t-d_2\right)\right)-S_1\left(t\right)\cdot R_A$

$\begin{array}{c}{v}_2\frac{\text{d}{S}_2\left(t\right)}{\text{d}t}=\left|q_1\right|\cdot \left(S_1\left(t-d_2\right)-S_2\left(t-d_2\right)\right)\\ +\left|q_2\right|\cdot \left(S_3\left(t-d_2\right)-S_2\left(t-d_2\right)\right)-S_2\left(t\right)\cdot R_N-F_s\end{array}$

$v_3\frac{\text{d}{S}_3\left(t\right)}{\text{d}t}=\left|q_2\right|\cdot \left(S_2\left(t-d_2\right)-S_3\left(t-d_2\right)\right)+F_s$

In addition, we consider temperature as dynamic variables in our model framework. Assuming the ambient temperature for Arctic, high-latitude North Atlantic and low-latitude North Atlantic are $c_1,c_2,c_3$, and let $r_1,r_2,r_3$ characterizing the speed of heat conducting into the water, we derive:

$v_1\frac{\text{d}{T}_1\left(t\right)}{\text{d}t}=\left|q_1\right|\cdot \left(T_2\left(t-d_2\right)-T_1\left(t-d_2\right)\right)-T_1\left(t\right)\cdot R_A+r_1\left(c_1-T_1\left(t\right)\right)$

$\begin{array}{c}{v}_2\frac{\text{d}{T}_2\left(t\right)}{\text{d}t}=\left|q_1\right|\cdot \left(T_1\left(t-d_2\right)-T_2\left(t-d_2\right)\right)+\left|q_2\right|\cdot \left(T_3\left(t-d_2\right)-T_2\left(t-d_2\right)\right)\\ -T_2\left(t\right)\cdot R_N+r_2\left(c_2-T_2\left(t\right)\right)\end{array}$

$v_3\frac{\text{d}{T}_3\left(t\right)}{\text{d}t}=\left|q_2\right|\cdot \left(T_2\left(t-d_2\right)-T_3\left(t-d_2\right)\right)+r_3\left(c_3-T_3\left(t\right)\right)$

The delayed feedback mechanisms within the Arctic climate system exhibit unique temporal characteristics during abrupt climate transitions. When the AMOC intensifies, warm water reaches the Arctic marine margins within 1 - 3 years [42] . This rapid advection stems from the Last Glacial Maximum paleogeographic configuration: lower sea levels extended ice sheets to continental shelf edges, enabling direct thermal erosion of ice-shelf basal zones—a stark contrast to modern fjord-dominated retreat [43] .

Arctic ice sheets respond to oceanic forcing through three distinct stages:

1) Initial basal melting (0 - 15 years): Subsurface warming triggers ice-shelf thinning at rates > 10 m/y, recorded by abrupt increases in ice-rafted debris with Laurentide-sourced dolomite in North Atlantic sediments [44] ;

2) Ice dynamic acceleration (15 - 30 years): Reduced buttressing causes inland ice-stream acceleration, evidenced by layer-thinning in Greenland ice cores [45] ;

3) Runoff peak (30 - 50 years): Combined meltwater discharge and iceberg calving release rates exceeded 0.1 Sv freshwater, as reported in [46] [47] .

In addition, the accumulation of freshwater is subject to complex spatiotemporal patterns, with proglacial lakes attaining outburst capacities (up to 500 Gt) in 50 - 150 years, as seen in varved sediment records from Lake Agassiz [48] , and subglacial aquifers storing freshwater on time scales of 200 to 500 years, as evidenced by porewater Cl⁻ profiles and ³⁶Cl dating [49] .

Despite decadal-scale freshwater discharge into the ocean, substantial weakening of the AMOC manifests after approximately 250 - 350 years. This delayed response primarily stems from time-dependent accumulation processes in critical convection zones. Norwegian Sea sediment Nd records reveal a 260 ± 40 yr delay between ice-rafted debris events and subsequent AMOC collapse [50] .

There is a high-resolution archive of documents in lag hierarchy as follows. Antarctic EPICA ice core δ18O excursions lag Greenland stadial—interstadial transitions by 310 ± 50 years on average [51] . South Atlantic Nd data further indicate that AMOC signal propagation via oceanic pathways after approximately 190 ± 30 years, giving rise to a bipolar see-saw lag of 300 ± 70 years [52] .

By integrating all feedback stages into one sequence of assessment, we suggest a delay of 400 years as a very conservative estimate representing a robust time-lag to ice-sheet-ocean-climate interaction throughout the system.

Self-sustained multidecadal AMOC oscillations are obtained in the revised three-box Stommel model. Figure 11 shows the numerical result at $d_1=400y,d_2=20y$.

First, let’s examine the temperature variations. When the strong AMOC initiates, the high-latitude North Atlantic and Arctic regions experience abrupt warming, with the former showing a rapid temperature increase of approximately 6˚C and the latter rising by about 4˚C within a short timeframe, which is consistent with the results shown in [35] . Following the collapse of the strong AMOC, temperatures in both regions gradually decrease and eventually return to their initial levels.

And from the graphs, we can observe that a Dansgaard-Oeschger (D-O) cycle consists of the following phases:

1) At point a (see Figure 12 ), as freshwater forcing ( $R_N,R_A$) declines to a critical threshold, it triggers the activation of a strong AMOC.

2) Point b marks the peak intensity of the strong AMOC. After about 400 years, under persistent ice sheet runoff from Arctic ice sheets (primarily meltwater input), freshwater forcing increases, leading to the collapse of the strong AMOC and its transition to a weak mode. Point c denotes the terminus of the strong AMOC (Strong phase).

3) From point d to e, under weak AMOC conditions, ice sheet runoff diminishes, and freshwater forcing gradually decreases. During this period, freshwater forcing declines to negative values, leaving the system in a monostable state dominated solely by the weak AMOC. Notably, the AMOC intensity increases with periodic fluctuations. This pattern primarily arises because when Arctic runoff (R) remains stable, the AMOC can reach its equilibrium state within a timescale that is a bit shorter than the characteristic temporal lag of the system. The cumulative effects of AMOC’s influence on Arctic runoff only become manifest after the subsequent 400-year phase. This delayed feedback triggers freshwater forcing adjustments, driving the AMOC toward a new equilibrium state.

4) At point e, freshwater forcing declines to negative value which triggers the next cycle.

All the cycles go through the same sequence and each lasts around 1500 - 2000 years, which is about 4 - 5 times the typical time delay of the entire system. The cycle duration largely relies on Phase C—the most time-consuming phase—lasting for a very long period of time.

In our model, the influence of AMOC (Atlantic Meridional Overturning Circulation) on ice-sheet runoff only becomes evident after 400 years. This temporal lag allows the AMOC to reach its new steady state under the ice-sheet runoff values from the previous phase, consequently causing the AMOC variation to demonstrate this pattern. Therefore, we can approximately discretize the model

regarding time t into a recursive sequence(see Figure 13 ). Strong AMOC can only restart when freshwater forcing drops below zero. Consequently, the timing of strong AMOC restart depends on which term in this sequence first attains negative freshwater forcing values. Figure 13 illustrates this process.