A minimal model for wind- and mixing-driven overturning: threshold behavior for both driving mechanisms

Abstract

We present a minimal conceptual model for the Atlantic meridional overturning circulation which incorporates the advection of salinity and the basic dynamics of the oceanic pycnocline. Four tracer transport processes following Gnanadesikan in Science 283(5410):2077–2079, (1999) allow for a dynamical adjustment of the oceanic pycnocline which defines the vertical extent of a mid-latitudinal box. At the same time the model captures the salt-advection feedback (Stommel in Tellus 13(2):224–230, (1961)). Due to its simplicity the model can be solved analytically in the purely wind- and purely mixing-driven cases. We find the possibility of abrupt transition in response to surface freshwater forcing in both cases even though the circulations are very different in physics and geometry. This analytical approach also provides expressions for the critical freshwater input marking the change in the dynamics of the system. Our analysis shows that including the pycnocline dynamics in a salt-advection model causes a decrease in the freshwater sensitivity of its northern sinking up to a threshold at which the circulation breaks down. Compared to previous studies the model is restricted to the essential ingredients. Still, it exhibits a rich behavior which reaches beyond the scope of this study and might be used as a paradigm for the qualitative behaviour of the Atlantic overturning in the discussion of driving mechanisms.

Appendices

Appendix 1: Scaling of the northern sinking

This section briefly presents five different approaches to formulate scaling laws for the northern sinking. For each approach, a short derivation is given, which identifies the main assumptions and discusses their validity. In this context, the difference between two vertical scales is emphasised, the pycnocline depth D and the level of no motion $\Uplambda$. The first description from Robinson (1960), also called the classical scaling, assumes the meridional overturning circulation to be in geostrophic balance. The vertical derivative of the momentum equation in the steady state yields

$$ f \frac{\partial v}{\partial z} = - g \cdot \frac{\partial \rho}{\partial x}, $$

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when the hydrostatic equation is applied. Here v denotes the meridional velocity, ρ is the ocean density field and g is the gravitational constant.

Two scales, one for the meridional velocity field V and one for the characterisitic depth $\Uplambda$ of the vertical profile of horizontal velocities, are introduced. The second is identified with the level of no motion, where the mean meridional velocities vanish. Additionally using a scale for the zonal density gradient $\Updelta_x \rho$ occurring over a length scale L _x gives

$$ V = \frac{g}{f}\frac{\Updelta_x \rho}{\rho_0}\cdot \frac{\Uplambda}{L_x}. $$

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In order to transform the zonal density gradient into a meridional one $\Updelta_y \rho$, the zonal and meridional velocity scales are linked. Albeit a radical generalisation, a constant ratio V = C _cl U is assumed.

$$ V = C_{cl} \frac{g}{f} \frac{\Updelta_y \rho}{\rho_0}\cdot \frac{\Uplambda}{L_y} $$

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Since we seek an expression for the Atlantic overturning, an integration over the zonal extent L _m and the vertical extent $\Uplambda$ of the flow is conducted. This gives the classical scaling for the northern sinking

$$ m_N = V \cdot \Uplambda \cdot L_m = C_{cl} \frac{g} {f}\frac{\Updelta_y \rho}{\rho_0}\frac{L_m}{L_y}\cdot \Uplambda^2, $$

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which is proportional to the meridional density gradient, to the ratio between zonal and meridional length scales $L_m / L_y$ and to the square of the level of no motion $\Uplambda$. The theoretical basis for this estimate constrains its spatial applicability to (1) the geostrophic assumption which is not valid near continental boundaries and to (2) the ad-hoc transformation from zonal to meridional density gradients. Gnanadesikan (1999) suggests that a western boundary current exerts control on the northern sinking. Neglecting the velocity component perpendicular to the boundary, a relation between the meridional pressure gradient and the meridional velocity is derived

$$ \frac{1}{\rho_0} \frac{\partial p}{\partial y} = \nu \nabla^2 v = \nu \frac{\partial^2 v}{\partial x^2} , $$

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where ∇² is the Laplacian and ν is the dynamic viscosity in the boundary current. For the second equality it is assumed that the zonal change in meridional velocity exceeds the changes in vertical and meridional direction by several orders of magnitude (Montoya et al. 2005).

A scale analysis analogue to the previous paragraph provides

$$ \nu \frac{V}{L_m^2} = C_{G99} \frac{1}{\rho_0} \frac{\Updelta_y p} {L_y} = - C_{G99} g \frac{\Updelta_y \rho}{\rho_0} \frac{D}{L_y}, $$

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where the constant C _G99 accounts for any effects of geometry and boundary layer structure. For the second equality, where the hydrostatic equation is employed, a new vertical scale height D is introduced. This height represents the density stratification of the ocean and is referred to as the pycnocline depth D.

Integration yields another scaling law for the northern sinking

$$ m_N = V \cdot \Uplambda \cdot L_m = C_{G99} \frac{g L_m^2}{\nu} \frac{\Updelta_y \rho}{\rho_0}\frac{L_m}{L_y} D \Uplambda . $$

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In analogy with the classical scaling law, we again find proportionality to a meridional density difference. However the relation between the Coriolis frequency f is replaced by the inverse of the zonal viscosity time scale in the western boundary current L ² _m /ν. Another even more important change is the proportionality to the product of the pycnocline depth D and the level of no motion $\Uplambda$. This approach is drawn from the assumption that a meridional pressure gradient causes a frictional western boundary current which limits the deep water formation in the Nordic Seas. In contrast to this, Johnson and Marshall (2002) base their scaling of the northern sinking on an ocean model of reduced gravity. It is built up by a surface layer of depth h and an infinitely deep and motionless lower layer of fixed density. In this set-up, the level of no motion is implicitly equal to the pycnocline depth. Assuming a geostrophic flow in the interior of the surface ocean basin to provide the water needed for the northern sinking m _N , the meridional velocity becomes

$$ v = -\frac{g}{f} \frac{\Updelta_z \rho}{\rho_0} \cdot \frac{\partial h}{\partial x}, $$

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where $\Updelta_z \rho$ is the vertical density difference between the two layers. Since the depth h is a function of x, a zonal integration from the western to the eastern boundary leads to

$$ m_N = \int\limits_{x_w}^{x_e} h v dx = \frac{g \Updelta_z \rho}{2 f \rho_0} \cdot \left( h_E^2 - h_W^2 \right) $$

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$$ = \frac{g \Updelta_z \rho}{2 f \rho_0} \cdot \tilde{D}^2 , $$

(37)

where $h_E, h_W$ are the layer depth at the eastern and western boundary, respectively. The third step implies Johnson’s redefinition of the pycnocline depth $\tilde{D}$ (Johnson et al. 2007).

The main difference to previous scalings is a dependence on a density gradient $\Updelta_z \rho$ in the vertical direction. In addition this approach sees the reason for the geostrophic flow in a zonal tilt of the pycnocline depth. However, it is argued that an outcropping of the pycnocline occurs at the western boundary, while at the eastern boundary the pycnocline is equal to D (Johnson et al. 2007). A fundamentally different approach is provided by Guan and Huang (2008) who introduce an energy constraint, instead of the well known buoyancy constraint. The idea is that the energy supply is used for diapycnal mixing, which is described by a vertical advection-diffusion balance. Using a scale for the vertical density difference $\Updelta_z\rho$, the scale of its vertical change $\Updelta_z(\Updelta_z\rho)$ and another for the pycnocline depth D, the equation can be rewritten for a constant diapycnal diffusivity κ

$$ w \Updelta_z\rho = \frac{\kappa}{D} \Updelta_z(\Updelta_z\rho). $$

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Assuming an exponential density profile, the proportionality $\Updelta_z(\Updelta_z\rho) \sim \Updelta_z\rho$ becomes valid.

The gravitational potential energy (GPE) in a two-layer box model, with a vertical density difference $\Updelta_z\rho$, increases due to vertical mixing with a rate of $-g \kappa\Updelta_z\rho$ (per unit area). Thus, meridional and zonal integration over a range of respectively L _m and B yields

$$ E_{pot} = - g \Updelta_z\rho \kappa L_m B = - g \Updelta_z\rho w D L_m B . $$

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Rearranging the equation and integrating over the same horizontal plane, a new scaling of the northern sinking arises

$$ m_N = -\frac{E_{pot}}{g \Updelta_z\rho D}. $$

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It is obvious that this approach diametrically opposes the ones above, because the northern sinking m _N is now inversely proportional to the vertical density gradient $\Updelta_z\rho$ and D. This discrepancy is caused by its fundamentally different assumptions. Unlike in the others, an energy source for the circulation is included, which maintains diapycnal diffusion and therefore produces available potential energy E _pot for the northern sinking. Marotzke (1997) introduces a scaling law similar to the classical approach. The difference lies in a convincing transformation from a zonal density gradient into a meridional one. Marotzke links them by using several assumptions about the density stratification of the ocean:

• 1. The density of the ocean surface is solely a linear function of latitude and the properties of the deep ocean are given by the surface water of highest density.

• 2. The density of the ocean surface is solely a linear function of latitude and the properties of the deep ocean are given by the surface water of highest density.

• 3. The occurrence of Kelvin and Rossby waves in equatorial regions eliminates all zonal isopycnal slopes except for the western boundary current.

• 4. At the eastern boundary, a well mixed surface layer down to a fixed depth z _ρ is assumed. This depth is prescribed in equatorial regions up to a specific latitude, where it becomes zero. This means that the isopycnal that separates the surface layer from the abyssal ocean, outcrops at a defined latitude.

This idealised stratification provides a linear relation between zonal and meridional density differences

$$ \Updelta_x \rho(y,z_\rho) = \Updelta_y \rho(y,z_\rho) \left( 1 - y/L_y \right) \cdot y/L_y , $$

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calculated at a specific depth z _ρ, with L _y being the meridional extent of the basin. The density differences $\Updelta_x \rho, \Updelta_y \rho$ are determined at opposite edges of the North Atlantic basin, respectively in meridional and zonal directions. Since the dependence on latitude is not the main focus here, the latitudinal maximum for the equation is used, which is attained at $y = \frac{1}{2} L_y$. Inserting in the meridional geostrophic equation and applying a scale analysis yields

$$ \frac{V}{\Uplambda} = C_{Ma} \frac{g}{f} \frac{\Updelta_y \rho} {\rho_0} \frac{1}{L_x}, $$

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where C _Ma is a constant accounting for geometry.

Integrating twice in vertical direction from the surface to the level of no motion gives

$$ m_N = C_{Ma} \frac{g}{f} \frac{\Updelta_y \rho}{\rho_0} \frac{L_m}{L_x} \cdot \Uplambda^2 . $$

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In correspondence to the classical scaling law, this one also predicts a proportionality of the northern sinking to a meridional density difference $\Updelta_y \rho$ of the ocean surface layer, and to the square of the level of no motion $\Uplambda$. Apart from geostrophy, it is the four assumptions from above which should be evaluated to judge the validity of this scaling. Marotzke argues that the first three have already been used successfully and therefore are generally accepted. The last one for the eastern boundary layer is based on model observations (Marotzke 1997).

In this work, the decision fell on the approach of Marotzke (1997), because it combines the idea of a geostrophic current and a boundary layer theory. In this way, the zonal pressure gradient is convincingly converted into a meridional one. Note that although the level of no motion $\Uplambda$ and the pycnocline depth D are physically different, these two scales cannot be seen as independent from each other. The level of no motion definitely separates two water bodies whose dynamics brings water from spatially separated areas. But this also creates a significant difference in the salinity and temperature characteristics, which gives rise to strong stratification.

Appendix 2: Discriminants

This section deals with the derivation of the discriminants for the various model subcases. They contain all information needed to characterise the transition of the model between different dynamic regimes.

A polynomial p _n of degree n in one variable $x \in \mathcal{C}$ is described via its roots

$$ p_n(x) = a_n \prod_{i=1}^{n} \left( x - \alpha_i \right) = \sum_{i=0}^{n} a_i \cdot x^i, $$

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where $\alpha_1, \alpha_2, ..., \alpha_n$ are the roots of p _n and a _n is the coefficient of the highest order term. One general form to determine its discriminant is

$$ \Upupsilon[p_n] := n^n (-1)^{\frac{n(n-1)}{2}} a_n^{2n-2} \prod_{i<j} \left( \alpha_i - \alpha_j \right)^2 . $$

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Given a concrete polynomial p _n (x) with a _n  ≠ 0, then $\Upupsilon[p_n] = 0$ if and only if p _n has a double root [Mathematics: Theory & Applications, Discriminants, Resultants and multidimensional Determinants, p.404,6]. This can directly be deduced by applying the definition of the discriminant. It implies that a change in the amount of real roots is indicated by the roots of the discriminant, which are a function of the coefficients a _i . If the sign of the discriminant $\Upupsilon[p_n]$ changes, it has drastic implications for the roots of the polynomial p _n . This is best illustrated by a quadratic polynomial p ₂, whose roots are found at

$$ \alpha_{1/2} = \frac{-a_1 \pm \sqrt{a_1^2 - 4 a_2 a_0}}{2 a_2}. $$

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The discriminant is the negative term within the square root $\Upupsilon[p_2] = 4 a_2 a_0 - a_1^2$ and its sign determines whether the two roots are imaginary or real. For cubic polynomials p ₃, $\Upupsilon[p_3] < 0$ signifies three real roots, $\Upupsilon[p_3] = 0$ one real root and $\Upupsilon[p_3] > 0$ two imaginary and one real root. For polynomials of degree $n \geq 4$, the connection between the sign of the discriminant and the characteristics of the roots becomes more elaborate and will not be used later.

In general, we assume that all volume flux constants $C_N, C_U, C_W, C_E$, the mean density ρ, the average salinity S ₀, the expansion coefficients $\alpha_T, \beta_S$, the freshwater bridges $F_N, F_S$ and the pycnocline depth D are positive. In addition the pole to equator temperature difference $\Updelta\theta$ can only be negative. This defines the physical parameter space, which we are going to adopt in the following discussion.

1.1 2.1 Mixing-driven case

Instead of explicitly calculating the discriminant of the governing polynomial of the mixing-driven case (eq. (13)), a mathematical tool from number theory has been used (Leutbecher 1996, p.225), since it provides a simple way to compute it. A so-called resolvent function is determined by the original polynomial of degree four. On one hand, this new function is a polynomial whose order is reduced by one and its discriminant equals that of the governing polynomial (eq. (13)). It allows thus to calculate the discriminant via a polynomial of degree 3, which yields

$$ \Upupsilon_M = 256 C_N^3 \rho_0^3 \beta_S^3 S_0^3 F_N^3 C_U^6 - 27 C_N^4 \rho_0^4 C_U^8 \alpha_T^4 \Updelta \theta^4. $$

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Its zero transitions determine a critical value, where the dynamic of the model changes. The root of the discriminant reads, with respect to the freshwater flux,

$$ F_M^* = \frac{3 \left(2 C_N \rho_0\right)^{1/3} C_U^{2/3} \alpha_T^{4/3} }{8 \beta_S S_0} \left|\Updelta\theta\right|^{4/3}. $$

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In order to compute the critical pycnocline depth and the corresponding northern sinking, an additional equation for the critical value is needed. A lemma from Galois theory states that the discriminant of a classical polynomial p _n (x) in one variable x can be determined via the first derivative of the polynomial p _n ′(x). We have

$$ \Upupsilon[p] = n^n a_n^{n-1} \prod_{\gamma: p_n'(\gamma) = 0} p_n(\gamma), $$

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where n is the maximal order of the polynomial and a _n the coefficient of the highest order term (Gelfand et al. 1994, p.404). If the discriminant is zero for a critical choice of coefficients, at least one root of the polynomial p _n coincides with one root of its first derivative p _n ′.

The derivative of the governing polynomial of the mixing case (eq. (13)) with respect to F _N yields

$$ \frac{\partial D}{\partial F_N} = - \frac{\beta_S S_0 D^4}{4 \beta_S S_0 F_N D^3 + 3 \alpha_T \Updelta\theta C_U D^2} , $$

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while the denominator is the derivative of the same polynomial with respect to D. Using the lemma, the discriminant vanishes at the critical point F ^* _M and, consequently, the first D-derivative of the polynomial shows a root. Since this derivative appears in the denominator of ∂D/∂F _N , it diverges at F ^* _M . In addition, there is only one real value for F _N where the denominator crosses zero and thus the derivative ∂D/∂F _N is positive for physical D as long as the $F_N < F_M^*$. Finally, by inserting the critical value F ^* _M into the derivative of the polynomial with respect to D, an additional equation is obtained that permits the determination of the critical pycnocline depth D ^* _M

$$ D_M^* = \left( \frac{4 C_U}{C_N \rho_0 \alpha_T \left| \Updelta\theta \right|} \right)^{1/3}. $$

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The relation $m_N = \frac{C_U}{D}$ provides the key to find the critical northern sinking.

1.2 2.2 Wind-driven case

The discriminant for the governing equation of the wind-driven case (eq. (20)) is calculated in a straightforward manner (Gelfand et al. 1994, p.405)

$$ \begin{aligned} \Upupsilon_W &= 4 \left\{ C_N^3 \rho_0^3 C_W^5 \alpha_T^3 \Updelta \theta^3\right.\\ & \quad +\left( -\frac{1}{4} C_N^2\rho_0^2 C_E^2 C_W^4 + 3 C_N^3 \rho_0^3 C_W^4 \beta_S S_0 F_N \right) \alpha_T^2 \Updelta \theta^2 \\ &\quad + \left( 3 C_N^3 \rho_0^3 C_W^3 \beta_S^2 S_0^2 F_N^2 - 5 C_N^2\rho_0^2 C_E^2 C_W^3 \beta_S S_0 F_N\right) \alpha_T \Updelta \theta \\ &\quad + \left.C_N \rho_0 C_E^4 C_W^2 \beta_S S_0 F_N + 2 C_N^2\rho_0^2 C_E^2 C_W^2 \beta_S^2 S_0^2 F_N^2\,+\,C_N^3 \rho_0^3 C_W^2 \beta_S^3 S_0^3 F_N^3 \right\} \end{aligned} $$

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This is a third order polynomial in F _N and, in general, there are three roots that determine the critical freshwater input. For our parameter set, only one real root can be found, which we refer to as F ^* _W . In general, F ^* _W denotes the largest possible critical freshwater input. Although its solution can be analytically determined, it shows a lack of lucidity and therefore an approximation is presented. It is possible to deduce that the derivative of F ^* _W with respect to $\Updelta \theta$ is constant in the limit $\Updelta \theta \longrightarrow -\infty$. To show this, the discriminant $\Upupsilon_W$ is divided by $\Updelta\theta^3$. Considering both the temperature difference limit and that $\Upupsilon_W$ vanishes at the critical point, yield that only a linear term remains. Without the offset, the linear approximation reads

$$ \left( F_W^* \right)_{appr} \approx - \frac{C_W}{\beta_S S_0} \alpha_T \Updelta \theta. $$

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It is the temperature difference between the North Atlantic and the equator in combination with the SO winds that create a non-zero critical freshwater flux. The linear estimate for F ^* _W approximates the slope of the analytic solution fairly well as long as

$$ \left| \Updelta\theta \right| \gg \frac{C_E^2}{C_N \rho_0 C_W \alpha_T}= 0.5^{\circ}C. $$

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This expression is deduced from $\partial F_W^* /\ {\partial \Updelta\theta}$, which is obtained by the $\Updelta\theta$-derivative of the discriminant at F ^* _W . Knowing that the maximal dependence of F ^* _W on $\Updelta\theta$ is linear (fact of the approximation), a comparison of the terms in the resulting expression gives this restriction to the applicability of the linearisation.

Besides, using the same line of argument as in the previous section (Gelfand et al. 1994, p.404), the D-derivative of the governing polynomial of the wind-driven case (eq. (20)) is zero at F ^* _W . This provides, on one hand, the information that ∂D / ∂F _N is positive for $F_N < F_W^*$ (if there are more than one real roots for F ^* _W choose the smallest), and on the other, it provides an additional equation to determine the critical pycnocline depth. Inserting the linear approximation for F ^* _W yields

$$ D_W^* \approx \frac{C_E^2 - \sqrt{C_E^4 - 6 C_N \rho_0 C_E^2 C_W \alpha_T \Updelta\theta}}{3 C_N \rho_0 C_E \alpha_T \Updelta\theta}. $$

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The linear approximation without the offset $\left( F_W^* \right)_{appr}$ exhibits an additional feature: it serves as an upper boundary for F ^* _W for negative $\Updelta\theta$. The complete analytic solution for F ^* _W as the root of $\Upupsilon_W$ has the following structure

$$ \begin{aligned} F_W^* &= \xi \cdot\Updelta\theta + \eta + \left\{ - \zeta(\Updelta\theta) + \sqrt{\chi(\Updelta\theta)} \right\}^{1/3} &\quad - \left\{ \zeta(\Updelta\theta) + \sqrt{\chi(\Updelta\theta)} \right\}^{1/3}. \end{aligned} $$

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Since ξ is the same constant which was already found in our approximation, the other terms must sum up to a negative value, if the approximation should serve as an upper boundary. The respective terms are calculated via

$$ \begin{aligned} \eta &= -\frac{2}{3}\cdot\frac{C_E^2}{C_N \rho_0\beta_S S_0} \\ \zeta &= \zeta_2 \cdot \Updelta\theta^2 + \zeta_1 \cdot \Updelta\theta - \zeta_0\\ \chi &= \chi_4 \cdot \Updelta\theta^4 - \chi_3 \cdot \Updelta\theta^3 + \chi_2 \cdot \Updelta\theta^2 - \chi_1 \cdot \Updelta\theta, \end{aligned} $$

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which implies that η is negative. Furthermore, χ is positive for all $\Updelta\theta \leq 0$, because the constants χ _i are all positive and terms with odd exponents are without exception multiplied by −1. Thus, if $\zeta$ would be positive then the sum of the two cube roots is negative, because one substracts the root of the sum of $\zeta$ and $\sqrt{\chi}$ from the root of their difference. However, $\zeta$ is a polynomial of degree two with exclusively positive constants $\zeta_i$ with zeros at

$$ \Updelta\theta_{1/2} = \left( -\frac{10}{27} \pm \frac{2\sqrt{3}}{9} \right) \cdot \frac{C_E^2}{C_N \rho_0 C_W \alpha_T}. $$

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For our parameters, this implies that below a value of $\Updelta\theta = -0.4 ^{\circ}C$ our approximation provides an upper boundary for the critical freshwater flux. Observed ocean temperature differences between the North Atlantic and the equator $\Updelta\theta$ differ considerably from zero and are likely on the order of several degrees.

1.3 2.3 Full problem

For the full problem, the discriminant for the normalised form of the governing eq. (11) is

$$ \begin{aligned} \Upupsilon_F &= a^4 \left( - 27 d^4 + 144 c d^2 e - 128 c^2 e^2 - 192 b d e^2\right) \\ &\quad + 2a^3 \left( -2c^3 d^2 + 8 c^4 e - 40 b c^2 d e - 3 b^2 d^2 e\right.\\ &\quad - \left.18 d^3 e - 800 b e^3 + 9 b c d^3 + 72 b^2 c e^2 + 80 c d e^2 \right)\\ &\quad + a^2 \left( -6 c^2 d^3 + 24 c^3 d e -27 b^4 e^2 - 50 d^2 e^2\right.\\ &\quad + 2000 c e^3 - 4 b^3 d^3 + 18 b^3 c d e + 144 b d^4\\ &\quad - 746 b c d^2 e + 560 b c^2 e^2 + b^2 c^2 d^2 - 4 b^2 c^3 e\\ &\quad + \left. 1020 b^2 d e^2 \right) + a \left( 24 b^3 d^2 e\right.\\ &\quad - 630 b^3 c e^2 + 18 b c^3 d^2 - 72 b c^4 e + 160 b d^3 e \\ &\quad - 2050 b c d e^2 - 80 b^2 c d^3 + 356 b^2 c^2 d e\\ &\quad - 2250 b^2 e^3 - 192 c d^4 + 1020 c^2 d^2 e - 900 c^3 e^2\\ &\quad - \left. 2500 d e^3 \right) + \left( -27 c^4 d^2 + 256 d^5\right.\\ &\quad + 108 c^5 e - 1600 c d^3 e + 108 b^5 e^2 + 2250 c^2 d e^2\\ &\quad + 256 a^5 e^3 + 3125 e^4 + 16 b^4 d^3 - 72 b^4 c d e\\ &\quad + 144 b c^2 d^3 -630 b c^3 d e + 2000 b d^2 e^2 - 3750 b c e^3\\ &\quad - 128 b^2 d^4 + 560 b^2 c d^2 e + 825 b^2 c^2 e^2\\ &\quad - \left. 4 b^3 c^2 d^2 + 16 b^3 c^3 e - 900 b^3 d e^2 \right) , \\ \end{aligned} $$

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with the following definitions

$$ \begin{aligned} a &= - \frac{C_E^2 + C_N \rho_0 \alpha_T \Updelta \theta + C_N \rho_0 \beta_S S_0 F_N}{C_N \rho_0 C_E \alpha_T \Updelta \theta} \\ b &= - \frac{C_N \rho_0 C_U \alpha_T \Updelta \theta - 2 C_E C_W}{C_N \rho_0 C_E \alpha_T \Updelta \theta} \\ c &= - \frac{C_W^2 - 2 C_U CE}{C_N \rho_0 C_E \alpha_T \Updelta \theta} \\ d &= - \frac{2 C_U C_W}{C_N \rho_0 C_E \alpha_T \Updelta \theta} \\ c &= - \frac{C_U^2} {C_N \rho_0 C_E \alpha_T \Updelta \theta}. \\ \end{aligned} $$

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Analogous to the previous cases, the derivative of the polynomial (eq. (11)) is zero for the critical freshwater input (Gelfand et al. 1994, p.404). Although this provides the additional equation to determine the critical pycnocline depth, the required F ^* _F is not available in an analytic form (only one real F ^* _F is observed for our parameter set), as the discriminant $\Upupsilon_F$ is a polynomial of degree five in F _N . Moreover, such a polynomial has at least one real root.

Appendix 3: Sensitivity to F _N

The fundamental idea is to show that the Stommel model is more sensitive to a change in the freshwater flux F _N than our model. This is done by analysing the derivative of m _N with respect to F _N in the steady state. Again the set of parameters is chosen to be physical (see Appendix 2). The analysis, presented here, can be reduced to the following problem.

Given two functions

$$ \begin{aligned} f,g: {\mathcal{R}} \longmapsto & {\mathcal{R}} \\ f(x) = - \left( \frac{p}{q-x} \right)^{1/2} & g(x) = \frac{r}{x-s(x)} , \end{aligned} $$

with $x \in \mathcal{R}$ and $p,q,r \in \mathcal{R}^+$. The real function s is referred to as the pole function.

The intersections x _i of these two functions $f(x_i) = g(x_i)$ are determined via the following equation

$$ x_i = s(x_i) - \frac{r^2}{2p} \pm \frac{r}{p} \sqrt{p \cdot (q - s(x_i)) + \frac{r^2}{4}}. $$

If the expression within the root is negative, then there exist exclusively imaginary x _i . This implies that if the pole of function f at q has a smaller value than that of g at s(x), and their difference is larger than r ²/(4p), then the two functions do not cross for x < q. Assume q < s(x) − r ²/(4p), then

$$ g(x) \geq f(x) \quad \bigvee x \in {\mathcal{R}}: f(x) \leq 0, $$

The crucial question is which restriction for x is defined by the condition q < s(x) − r ²/(4p).

The derivative $\partial m_N / \partial F_N$ is chosen to measure the sensitivity of the various subcases and they are associated with f and g (x stands for F _N ). In the Stommel case, the analytic function for m _N is available (compare Sect. 3) and its derivative with respect to F _N is a function of the form f. On the other hand, we find for each subcase of our model an implicit form of the function $\partial m_N / \partial F_N$ which is represented by type g. Rearranging the specific volume flux balance (eq. (1)) gives an expression for m _N . The function $\partial m_N / \partial F_N$ is calculated using the F _N -derivative of the associated governing equation. The mathematical considerations reveal that the pole functions for all subcases of our model must exceed that of the Stommel case by more than r ²/(4p), in order to show that the Stommel (1961) model is more sensitive to F _N .

$$ F_{S}^{pole} = \frac{1}{4} \frac{\alpha_T \Updelta\theta} {\beta_S S_0} C_N \rho_0 \alpha_T \Updelta\theta D_S^2 \equiv F_S^* $$

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$$ F_{M}^{pole} = -\frac{3}{4} \frac{\alpha_T \Updelta\theta} {\beta_S S_0} m_U $$

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$$ F_{W}^{pole} = -\frac{1}{2} \frac{\alpha_T \Updelta\theta} {\beta_S S_0} \left( 2 m_W - 3 m_E \right) + \frac{m_E \left( m_W - m_E\right)}{C_N \rho_0 \beta_S S_0 D^2} $$

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$$ \begin{aligned} F_{F}^{pole} &= - \frac{\alpha_T \Updelta\theta}{\beta_S S_0} \left( m_W - \frac{5}{4} m_E + \frac{3}{4} m_U \right)\\ &\quad - \frac{m_E^2 - \frac{3}{2} m_E m_W + \frac{1}{2} m_W^2 - m_E m_U + \frac{1}{2} m_U m_W}{C_N \rho_0 \beta_S S_0 D^2}. \end{aligned} $$

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At first, the pole of the mixing-driven case F ^pole _M is compared to the constant one of the Stommel case F ^pole _S . Since the derivative of m _N with respect to F _N in the mixing-driven case is negative as long as F _N is smaller than F ^* _M (cf. function g), m _N decreases strictly monotonic with F _N (see Fig. 4). For $F_N = F_M^{pole}$, this derivative diverges and, consequently, the smallest value for m _N is reached. This minimal value equals the critical freshwater input F ^* _M (cf. Appendix 2) which provides an analytic expression. Note that solutions for $F_N > F_M^*$ are not physical (see Sect. 3.1). All in all, the position of the pole in the Stommel case F ^pole _S has to undercut the minimal value for F ^M _N corrected by the respective r ²/(4p) in order to be more sensitive. Since r ²/(4p) is in the order of 10⁻³ Sv, it will for the moment be neglected, to get a useful qualitative expression. We find an approximative upper boundary for Stommel’s prescribed pycnocline depth D _S

$$ D_S^{max} \approx \sqrt{3}\cdot\left( - \frac{C_U}{2 C_N \rho_0 \alpha_T \Updelta \theta} \right)^{1/3}, $$

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which has, in our framework (see Table 1), a value of 564.1 m. The exact solution can also be analytically derived giving constraints of $79.7 \,\hbox{m} \leq D_S \leq 558.4 \,\hbox{m} = D_S^{max}$. The lower analytic bound, is not physical because it causes a negative expression under the square root in eq. (8). The upper analytic bound is lower than our approximative value for all physical parameter sets and is in good agreement with the approximation. The analytic constraint is more stringent and therefore also guarantees that the Stommel (1961) model’s critical freshwater input F ^pole _S does not exceed that of our mixing-driven case F ^pole _M . As long as this criteria for the Stommel case is valid, it is sufficient to show that the poles for the wind-driven case F ^pole _W and the full problem F ^pole _F surpass the critical value of the mixing-driven case F ^* _M . Then Stommel’s model would be more sensitive to a change in freshwater flux F _N than our conceptual framework.

Starting with the wind-driven case, the condition $F_W^{pole} \geq F_M^*$ yields a polynomial of degree two

$$ \begin{aligned} 0 &\leq 6 \frac{\alpha_T \Updelta\theta}{\beta_S S_0} C_E \cdot D^2 + \frac{C_E C_W}{C_N \rho_0 \beta_S S_0} + \left\{ - \frac{C_E^2}{C_N \rho_0 \beta_S S_0}\right.\\ &\quad + \left.\frac{\alpha_T \Updelta\theta}{\beta_S S_0} \left(3 (\frac{1}{4} C_U^2 C_N \rho_0 \alpha_T \Updelta\theta)^{1/3} - 4 C_W \right) \right\}\cdot D, \end{aligned} $$

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whose two real roots have opposite signs for a physical choice of parameters. This conclusion can be made by comparing the signs of the different factors in the formula for the roots (not shown). The positive zero is an upper boundary to D in the wind-driven case D ^max _W  = 913 m. Figure 5a illustrates that for our parameter set, the stable branch of the pycnocline depth does not reach this critical value. The found constraint is merely implicit because all parameters must be set to determine the variable D. It actually indicates a direct restriction on the parameter space. A more rigorous approach makes use of the discriminant $\Upupsilon_W$ (eq. (53)). In order to check the sign of the discriminant, the freshwater flux F _N is substituted by the critical value of the mixing-driven case F ^* _M . As long as it is negative, three solutions for the wind-driven case exist. This is associated to a freshwater input, which is smaller than the maximal critical value for F ^* _W (see Appendix 2.2). The discriminant $\Upupsilon_W$ basically reads

$$ \Upupsilon_W = \sigma_4 \cdot C_E^4 + \sigma_2 \cdot C_E^2 + \sigma_0 , $$

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while σ _i are parameter dependent constants and σ₄ is positive (cf. Appendix 2.2). $\Upupsilon_W$ is a polynomial of degree four in C _E with even exponents which makes it axially symmetric around the ordinate. In general, four zeros are defined by

$$ (C_E)_{1/2/3/4} = \pm \left( \varepsilon_1 \cdot \left( \varepsilon_2 \pm \sqrt{\varepsilon_3}\right) \right)^{1/2}, $$

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with

$$ \begin{aligned} \varepsilon_1 &= \frac{C_N \rho_0}{2 \beta_S S_0 F_M^*} \\ \varepsilon_2 &= \frac{1}{4} C_W^2 \alpha_T^2 \Updelta\theta^2 + 5 C_W \alpha_T \Updelta\theta \beta_S S_0 F_M^* - 2 \beta_S^2 S_0^2 (F_M^*)^2\\ \varepsilon_3 &= \frac{1}{16} C_W^4 \alpha_T^4 \Updelta\theta^4 - \frac{3}{2} C_W^3 \alpha_T^3 \Updelta\theta^3 \beta_S S_0 F_M^* \\ &\quad + C_W^2 \alpha_T^2 \Updelta\theta^2 \beta_S^2 S_0^2 (F_M^*)^2 - 32 C_W \alpha_T \Updelta\theta \beta_S^3 S_0^3 (F_M^*)^3 . \\ \end{aligned} $$

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Excluding negative and imaginary values for C _E , only one positive zero remains, because $\varepsilon_1$ is positive and the difference between $\varepsilon_2$ and $\sqrt{\varepsilon_3}$ is negative. The second conclusion is based on the fact that from the first term in $\varepsilon_2$ something is subtracted, while to the first term in $\varepsilon_3$ something is added - as long as $\Updelta\theta$ is negative. The graph of $\Upupsilon_W$ has a lower boundary, because σ₄ is positive. Since for reasonable positive C _E only one zero exists, $\Upupsilon_W$ is negative up to this root $C_E^{max} = 1.2 \cdot 10^{-2} \hbox{Sv/m}$. This does not seem to be a severe criterion, because it lies one orders of magnitude above the used value (cf. Table 1). Under this condition, F ^* _W is bigger than that of the mixing-driven case F ^* _M . Since the function F ^pole _W is strictly monotonic decreasing with F _N (seen from its derivative), F ^pole _W is larger than the maximal F ^* _W . In other words $F_M^* \leq F_W^* \leq F_W^{pole}$. This ensures that the wind-driven case is overall less sensitive to a change in freshwater than Stommel (1961). To claim that the wind-driven case is less sensitive than the mixing-driven case, it is necessary to compare their two derivatives $\partial m_N / \partial F_N$ directly. For that the analytic solutions of the pycnocline depth in the two cases are needed, which would give a rather complex mathematical constraint. Therefore the analytic form of this constraint is omitted here but it is in principle depicted in Fig. 8. This figure indicates that for our parameter set the derivative in the mixing-driven case is higher than the one in the wind-driven case.

For the full problem an analytic constraint for the parameter set is not possible but an implicit one for the variable D can be provided. A polynomial of degree four arises from the condition $F_F^{pole} \geq F_M^*$

$$ \begin{aligned} 0 &\leq - D^4 \cdot 5 \mu C_E +\\ &\quad + D^3 \cdot \left( 4 \mu C_W - 4 F_M^* - 2 \nu C_E^2 \right) \\ &\quad + D^2 \cdot \left( 3 \mu C_U + 3 \cdot \nu C_E C_W \right) \\ &\quad + D \cdot \left(2 C_U C_E - C_W^2 \right)\nu \\ &\quad - \nu C_W C_U, \\ \end{aligned} $$

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with the positive constants

$$ \mu = - \frac{\alpha_T \Updelta\theta}{4 \beta_S S_0} $$

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$$ \nu = \left(2 C_N \rho_0 \beta_S S_0 \right)^{-1}. $$

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For our set of parameters, two positive real roots are found, along with a pair of negative roots. To ensure that the inequality holds, the positive roots are the upper and lower limit for the pycnocline depth of the full case $351 m \leq D \leq 1,329\,\hbox{m}$. The upper constraint is out of the range of the observed values, while the lower one is identified with a large negative freshwater input in the North Atlantic (cf. Fig. 7). If these constraints for the variable D are fulfilled, the pole function of the full problem F ^pole _F is smaller than the minimal one of the mixing-driven case F ^* _M . For the full problem to be less sensitive than the mixing-driven case additional constraints must hold. They emanate from the direct comparison of their respective derivatives $\partial m_N / \partial F_N$, which include the full analytic solutions for D in both subcases. We omit the complex, analytic form of the constraint since we find that our parameter set does not violate it (cf. Fig. 8).

Altogether, it can be concluded that if the Stommel (1961) model uses a pycnocline depth of $D_S \leq 558.4\,\hbox{m}$, then its northern sinking is more sensitive to a change in F _N than in our model. In other words, if the critical freshwater input of the Stommel model is slightly lower (10⁻³ Sv) than that of the mixing-driven case, then our model is less sensitive to a change in freshwater flux - if constraints on the parameter choice are considered. Under further conditions, it is argued that the mixing-driven case is less sensitive than the wind-driven case or even the full problem. All these constraints are fulfilled for our physical set of parameters.

Appendix 4: Salinity equations

Here we presents the equilibrium solutions for the box salinities with respect to the determined pycnocline depth D. All box salinities are given in reference to that of the northern box S _N . The system gets consistent if an average salinity for all boxes of S ₀ is prescribed, which gives the additional equation to determine S _N

$$ \left(V_N + V_U + V_D + V_S\right) \cdot S_0 = V_N \cdot S_N + V_U \cdot S_U + V_D \cdot S_D + V_S \cdot S_S . $$

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It is furthermore notable, that a direct influence of the entire model geometry is only noticed in the absolute salinities. Their relative differences are only dependent on D, which is a function of the width B and the meridional extent of the low-latitudinal box L _U . The relative values of the salinities are derived from the system of equations for the equilibrium salinities (see eq. (6)). A brief overview for the different subcases is given.

1.1 4.1 Mixing driven case

$$ S_U = S_N + \frac{F_N D}{C_U}S_0 $$

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$$ S_D = S_N $$

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In this special case the southern ocean box is disconnected from the others. Its salinity S _S is therefore independent and is only determined by the average ocean salinity.

1.2 4.2 Wind driven case

$$ S_U = S_N + \frac{F_N}{C_W - C_E D}\cdot S_0 $$

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$$ S_D = S_N + \frac{C_E D}{C_W}\left( \frac{F_N}{C_W - C_E D} - \frac{F_N + F_S}{C_W} \right)\cdot S_0 $$

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$$ S_S = S_N + \left( \frac{F_N}{C_W - C_E D} - \frac{F_N + F_S} {C_W} \right)\cdot S_0 $$

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1.3 4.3 Full problem

$$ S_U = S_N + \frac{F_N}{\frac{C_U}{D} + C_W - C_E D}\cdot S_0 $$

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$$ S_D = S_N + \frac{C_E D}{\frac{C_U}{D} + C_W - C_E D} \cdot \frac{C_E D F_N - \left( \frac{C_U}{D} + C_W - C_E D \right)F_S}{\frac{C_U}{D}C_W + C_W^2 + C_E C_U} \cdot S_0 $$

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$$ S_S = S_N + \frac{\frac{C_U}{D} + C_W}{\frac{C_U}{D} + C_W - C_E D} \cdot \frac{C_E D F_N - \left( \frac{C_U}{D} + C_W - C_E D \right) F_S}{\frac{C_U}{D}C_W + C_W^2 + C_E C_U} \cdot S_0 $$

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