## 1. Introduction

[2] The meridional overturning circulation in the Atlantic is a central challenge to our understanding of global climate dynamics. *Gnanadesikan* [1999] (hereinafter referred to as G99) presented a model for the deep meridional circulation in terms of the pycnocline depth (PD). This idealised model has been under intense investigation as a possible paradigm for the meridional overturning circulation [*Gnanadesikan and Hallberg*, 2000; *Saenko and Weaver*, 2003; *Gnanadesikan et al.*, 2002]. Furthermore it has been used to investigate the qualitative importance of different physical feedbacks on the oceanic circulation [*Klinger et al.*, 2003; *Gnanadesikan et al.*, 2003; *Kamenkovich and Sarachik*, 2004]. A qualitative feature of the deep meridional overturning circulation is the scaling relation between the volume transport and the meridional density difference in the Atlantic [*Bryan*, 1987]. Picking up Bryan's scaling arguments but assuming a constant PD in the Atlantic *Rahmstorf* [1996] proposed a linear relation which he demonstrated in the oceanic general circulation model (GCM) MOM-2. *Park* [1999] and *Scott et al.* [1999] derived the same scaling in a Stommel-type box model. GCM simulations of the ocean suggest that this linear relation carries over from the density to the pressure difference [*Hughes and Weaver*, 1994; *Thorpe et al.*, 2001]. The linear scaling relation between pressure difference and maximum overturning strength has since been demonstrated to be a robust feature in oceanic GCM simulations A. Griesel et al. (The role of eddies for the meridional pressure gradient and the strength of the Atlantic overturning circulation, manuscript in preparation, 2004; hereafter referred to as Gr04). In section 4 we present simulations with the coupled climate model CLIMBER-3α further supporting these findings.

[3] The G99 model contains four physical processes which influence the PD in the ocean. The balance of the pressure gradient in the NA and the frictional forces within the boundary currents leads to an equation for the northward volume transport

The pressure gradient is parameterised through the density difference in the NA Δρ, the north-south distance *L*_{y}^{(n)} over which the gradient occurs and the PD *D*.

The constant γ_{n} combines *L*_{y}^{(n)} with β, ρ and *C* (the meridional derivative of the Coriolis parameter *f*, the density and a proportionality constant of order one). *g* is the gravity constant. The quadratic dependence on *D* occurs due to the vertical integration in order to obtain a volume transport. In the SO the model includes the Drake passage effect through a wind-driven upwelling which does not explicitly depend on the PD *T*_{s}^{(e)} = (*L*_{x}τ)/(ρ*f*) ≡ 2γ_{e}. τ and *L*_{x} are the wind stress in the SO and the circumference around Earth at the latitude of Drake Passage. Additionally G99 includes an eddy induced return flow

where *v*_{ed} is the transport velocity which G99 parameterised following *Gent and McWilliams* [1990] while we focus here on its dependence on the PD. The fourth term in the model is associated with low-latitudinal upwelling described by an advection-diffusion balance *w*∂_{z}ρ = *K*_{v}∂_{zz}ρ in the tropics which yields

where *K*_{v} and *A*_{u} are the diapycnal diffusivity and the horizontal area of upwelling, respectively. All non-negative constants γ_{x} have been introduced for convenience. Note that the underlying assumption of the model is that these four process can be described using the same value *D* for the PD throughout the Atlantic. Equation (1) requires furthermore that the vertical extension of the northward volume flow is also given by *D*. Accepting these assumptions, the conservation of volume then results in the governing equation of the model

It can be shown that for all parameter settings the model has at most one solution with non-negative PD. In section 2 we give this solution analytically in terms of the volume transport *T*_{n} as a function of the pressure difference Δ*p* and discuss, in section 3, its scaling with Δ*p*. In section 4 we compare the results with simulations with the coupled climate model CLIMBER-3α.