Timescales of AMOC decline in response to fresh water forcing

Abstract

The Atlantic meridional overturning circulation (AMOC) is predicted to weaken over the coming century due to warming from greenhouse gases and increased input of fresh water into the North Atlantic, however there is considerable uncertainty as to the amount and rate of AMOC weakening. Understanding what controls the rate and timescale of AMOC weakening may help to reduce this uncertainty and hence reduce the uncertainty surrounding associated impacts. As a first step towards this we consider the timescales associated with weakening in response to idealized freshening scenarios. Here we explore timescales of AMOC weakening in response to a freshening of the North Atlantic in a suite of experiments with an eddy-permitting global climate model (GCM). When the rate of fresh water added to the North Atlantic is small (0.1 Sv; 1 Sv $=1\times 10^6$ m$^3$/s), the timescale of AMOC weakening depends mainly on the rate of fresh water input itself and can be longer than a century. When the rate of fresh water added is large ($\ge$ 0.3 Sv) however, the timescale is a few decades and is insensitive to the actual rate of fresh water input. This insensitivity is because with a greater rate of fresh water input the advective feedbacks become more important at exporting fresh anomalies, so the rate of freshening is similar. We find advective feedbacks from: an export of fresh anomalies by the mean flow; less volume import through the Bering Strait; a weakening AMOC transporting less subtropical water northwards; and anomalous subtropical circulations which amplify export of the fresh anomalies. This latter circulation change is driven itself by the presence of fresh anomalies exported from the subpolar gyre through geostrophy. This feedback has not been identified in previous model studies and when the rate of freshening is strong it is found to dominate the total export of fresh anomalies, and hence the timescale of AMOC decline. Although results may be model dependent, qualitatively similar mechanisms are also found in a single experiment with a different GCM.

Appendix: Salinity budget derivation

1.1 Model salinity budget

The salinity (S) evolution is controlled by the equation

$$\begin{aligned} \frac{dS}{dt} = -\nabla (S \mathbf {u}) + S(E-P) \delta (z- \eta ) + \mathcal {D} \end{aligned}$$

where $\mathbf {u}=(U,V,W)$ is the three dimensional velocity, E and P are the evaporation and precipitation respectively, and $\delta (z- \eta )$ is the Dirac function that is one at the surface ($z=\eta$) and zero elsewhere (Madec 2008). $\mathcal {D}$ contains diffusive and sea ice terms. Integrating over a volume of the Atlantic contained by land boundaries in the x direction, by latitudes $L_1$ and $L_2$ in the y direction and from the ocean floor ($z=-H$) to the surface $z=\eta$) we get

$$\begin{aligned} \int _{vol} \frac{dS}{dt} dxdydz =&\int _{L_1} VS dxdz - \int _{L_2} VS dxdz -\int _{surf} WS dxdy\\&+ \int _{surf} S(E-P) dxdy +\int _{vol} \mathcal {D} dxdydz. \end{aligned}$$

Since the model uses a linear free surface approximation (Roullet and Madec 2000; Madec 2008) the surface for the integrals are assumed to be at $z=0$ and hence the volume integrated over remains constant. This gives the approximation used by the model as

$$\begin{aligned} \frac{d}{dt}\int _{vol_0} S dxdydz =&\int _{L_1} VS dxdz - \int _{L_2} VS dxdz - \int _{z=0} WS dxdy \nonumber \\&+ \int _{z=0} S(E-P) dxdy +\int _{vol_0} \mathcal {D} dxdydz \end{aligned}$$

(7)

This gives the salinity budget in terms of the convergence due to lateral and vertical transports (with the vertical flux due to convergence in the free surface layer), surface fresh water fluxes (which change the salinity through implied changes in volume) and diffusive and ice terms. It is worth noting that the linear free surface has an impact on salt conservation: if the budget over the entire globe is considered, then all terms on the right hand side are zero apart from the third one. Although $\int _{z=0} W dxdy=0$ when considering a global integral, there can be a contribution from $\int _{z=0} WS dxdy$ although this is assumed to be small since W and S at the surface are generally not correlated.

Equation 7 can then be written as

$$\begin{aligned} \nu \frac{d\tilde{S}}{dt} = F_{adv} + F_{surf} + F_{D} \end{aligned}$$

(8)

where $\nu$ is the volume of the region, $\tilde{S}$ is the volume average salinity and $F_{adv}$ (first 3 terms on the right hand side of Eq. 7), $F_{surf}$ and $F_{D}$ are the advective fluxes, surface fluxes and diffusive terms respectively.

Now if we write $V=\overline{V} + v$ where $\overline{V} = \int V dx dz / A$ is the section mean velocity and $A= \int dx dz$ is the section area, then we can write

$$\begin{aligned} \int VS dx dz = \overline{V} ~\overline{S}A + \int v S dx dz \end{aligned}$$

where $\overline{S} = \int S dx dz / A$ is the section mean salinity. Similarly we can write

$$\begin{aligned} \int WS dx dy = \overline{W}^s ~\overline{S}^sA^s + \int w S dx dy \end{aligned}$$

where $\overline{W}^{s} = \int W dx dy / A^s$ is the area mean of the surface vertical velocity, $A^s= \int dx dy$ and $\overline{S}^s = \int S dx dz / A^s$.

Then

$$\begin{aligned} F_{adv}= F_{thr} + F_{S} +F_{N} + F_{ws} \end{aligned}$$

(9)

where

$$\begin{aligned} F_{thr} = -\overline{W}^s ~\overline{S}^sA^s +\overline{V_1} ~\overline{S_1}A_1 -\overline{V_2} ~\overline{S_2}A_2 \end{aligned}$$

is the advection due to the volume transport through the region having different salinities at the surface, northern and southern boundaries. We will refer to this as the throughflow component of the salinity transport.

$$\begin{aligned} F_{S}=\int v_1 S_1 dx dz, \end{aligned}$$

and

$$\begin{aligned} F_{N}=-\int v_2 S_2 dx dz \end{aligned}$$

are the exchanges of salt across the southern and northern boundaries respectively that are not part of the throughflow transport. The final term

$$\begin{aligned} F_{ws}= -\int w S dx dy \end{aligned}$$

is the covariance of vertical velocities and salinities. This term is small, but non-zero in the control, however remains constant throughout the hosing experiments. Hence the contribution to the salinity budget in Eq. 2 is negligable and the term is not included.

1.2 Advection decomposition

The advective terms in $F_S$ can be decomposed further in order to elucidate mechanisms. Note that although $F_N$ can be decomposed in the same way, we do not do this because $F_N$ is found to be small. One way of doing the decomposition is geometrically into zonal mean components and zonally varying anomalies. We write $v=v_{o}+v_{g}$ and $S=S_{o}+S_{g}$ where $a_{o} = \int a dx/\int dx$ so by definition $\int a_{g} dx=0$. Then

$$\begin{aligned} \int vS dx dz = \int v_{o}S_{o} dx dz + \int v_{g}S_{g} dx dz \end{aligned}$$

(10)

with the terms on the right often referred to as overturning ($F_{o}$) and gyre ($F_{g}$) components respectively.

Alternatively $F_S$ can be decomposed to identify whether advection changes because of circulation or salinity changes. Here we decompose $v = v_c + v'$ and $S=S_c + S'$ where $v_c$ and $S_c$ are time mean components from the control experiment. Then

$$\begin{aligned} \int vS dx dz = \int v_cS_c dx dz +\int v_cS' dx dz +\int v'S_c dx dz +\int v'S' dx dz. \end{aligned}$$

This is evaluated for both the hosing experiment and the control, and anomalies are taken with respect to time means of the control. Denoting the anomalous term with respect to the control using $\Delta \int vS dx dz$ we then have

$$\begin{aligned} \Delta \int vS dx dz = \Delta \int v_cS' dx dz + \Delta \int v'S_c dx dz + \Delta \int v'S' dx dz. \end{aligned}$$

(11)

The first term is the anomalous advection solely due to changing salinities (assuming that the circulation remains the same as in the control), $F_{v_cS'}$; the second term is the anomalous advection solely due to changing circulation (assuming that the salinity remains the same as in the control), $F_{v'S_c}$; the third term contains the anomalous advection due to co-varying salinity and circulation, $F_{v'S'}$. Note that the time mean of the latter term in the control can be non-zero if salinity and velocity co-vary due to internal variability.

1.3 Numerical calculation

To get an accurate calculation of Eq. 8, terms for the surface, diffusive and ice fluxes are calculated at every timestep in the model code and output as annual means. The tendency term is calculated as the salinity difference between monthly mean salinity fields at the beginning and end of each year. This is not an exact representation of the tendency, which should be calculated with instantaneous salinity fields, however errors are found to be small. The advective term $F_{adv}$ is calculated from the VS diagnostic where V and S are multiplied together (on the V grid) at every timestep within the model code. Unfortunately an error in the WS diagnostic meant that this had to be calculated from monthly mean W and S fields, however we believe that this has introduced little error. No terms were calculated as residuals. The sea ice terms were found to be negligible and the sum of the remaining terms were found to largely match the tendency terms.

To calculate the decompositions of $F_{adv}$ in Eq. 9, the monthly mean V and S fields were used. This means that $F_{adv}$ also includes an additional term of $F_{eddy}=\left\langle {VS}\right\rangle - \left\langle {V} \right\rangle \left\langle {S} \right\rangle$ where $\langle$ $\rangle$ indicates a monthly mean. Although this term is found to be important in other regions (Mecking et al. 2016), it is small across these boundaries so this term is neglected.