For this thesis a closed circuit (including coastline) of 224 station pairs was considered (Figure 3.1). These form four boxes, north, south, east and west, divided into 14 layers defined by isopycnal surfaces (Table 4.2). The layer divisions selected were based on the characteristic water masses of the Nordic Seas such that as much information was extracted from the system as possible (section!!5.2).
Full depth and layer-specific (all layers) conservation of volume and salinity (for salt) was required for each of the four boxes (north, south, east, west). The conservation of potential temperature (for heat) was also required in all layers that did not outcrop at the surface (i.e. layers 6 to 14), since heat can be exchanged with the atmosphere at the surface. Layer anomalies (for example, salinity minus the average salinity for the Nordic Seas) rather than absolute properties were used to reduce the dependence of the solution on the size of the units of the property equations.
To obtain an accurate set of reference velocities it is necessary to include some form of inter-layer exchange (Naveira Garabato et al., 2003; McIntosh and Rintoul, 1997). Here, diapycnal fluxes (volume, heat, salinity) were explicitly included for the relevant layer interfaces such that layer- specific conservation was physically consistent. The inclusion of independent diapycnal fluxes for each property represents the net diapycnal flux that results from all mixing processes which transfer mass, heat, or salinity between water masses in the ocean interior (Sloyan and Rintoul, 2001a). The vertical velocities and mixing were combined into an ‘effective’ diapycnal velocity which represents both advective and diffusive components:
†
w*=w+kC
∂C
∂z 4.20
where w* is the effective diapycnal velocity, w is the actual diapycnal velocity,
†
kC is the property diffusion coefficient and
†
∂C ∂z is the vertical gradient of the property C. The interfacial flux unknowns are here defined to have units of velocity and are referred to as effective interfacial velocities. For volume, the effective diapycnal velocity is the advective velocity, but for heat and salinity it has contributions from both advection and diffusion, and can be defined as the velocity that gives the total flux when multiplied by the interface area and mean property concentration. It is important to calculate separate effective transfer velocities for heat and salinity since there is asymmetry in the molecular behaviour of heat and salt (thermal diffusivity is 1.4!x!10-7!m2!s-1 and
the diffusivity of salt in water is 1.5!x!10-9!m2!s-1).
Chapter 4: Methods 83 † Dxj (vr+b)jCj dp - (wcA C)m+(wc A C)m+1 hm hm+1 Ú j=1 N Â +nc,m+1= 0 4.21
where j and m refer to the station pair and layer interface indices respectively, N is the total number of station pairs, Dx is the station spacing, C is the property concentration, vr is the baroclinic or
relative velocity, b is the barotropic velocity, wc is the effective interfacial velocity for each property
C, and A is the layer interface area within the domain. Thus for a layer, m, the equation represents the conservation of a property by summing around the sides of the layer (1st term), through the
bottom of the layer (2nd term) and through the top of the layer (3rd term). Noise in the constraint, i.e.
the extent to which exact conservation is not achieved, is represented by n. Noise is inherent in inverse problems involving the real ocean due in part to measurement problems with the data, and in part to the asynopticity of measurements.
Additional constraints are included to take into account prior knowledge of the circulation. Conservation of the volume transports within the deep basins below sill depth (~2800!m) is required in the Greenland and Lofoten Basins of the Greenland and Norwegian Seas. The volume flux between the Baltic and the rest of the Nordic Seas is constrained to zero (negligible effective volume exchange). Also, conservation of waters within the density range of Labrador Sea Water (36.8!<!s2!< 36.95) in the south box is required on the section between Iceland and Scotland, and
then over the remainder of the box, since the Iceland-Scotland Ridge effectively restricts exchange of these water masses within the domain of the box. Similarly the deep transports within the Rockall Trough (below 1200!m) are conserved, since the topography restricts communication of those waters with the rest of the south box.
The conservation equations were solved for a total of 364 unknowns. These comprised the 224 reference velocities (for each station pair) and the diapycnal fluxes for mass, salt (wm, ws; 13 layer
interfaces), and heat (wh; 8 layer interfaces) for each box. The resulting system of 165 equations and
364 unknowns was set up as a matrix equation after McIntosh and Rintoul (1997) and Wunsch (1978).
Ax = b 4.22
The (M x N) matrix, A, contains information about the geometry of the system; station pair layer areas multiplied by property concentrations, and layer interface areas multiplied by average interfacial property concentrations. The (1 x N) vector, x, contains the unknown reference velocities and the unknown interfacial ‘fluxes’ for mass, heat, and salt anomaly (wm, wh and ws respectively)
such that x = [v wm wh ws]T. The (M x 1) vector, b, contains information about the divergence due to the horizontal baroclinic property flux and the Ekman flux (i.e. the values to which the system is to be constrained). The system is underdetermined since there are fewer constraints (M) than unknowns (N), ie M < N.
Chapter 4: Methods 84
The inverse problem was solved using truncated SVD. Otherwise known as the ‘natural inverse’ this was introduced to solve such underdetermined systems by Lanczos (1961). Two sets of eigenvectors ui and vj may be found such that