369
high-latitude box and -0.2 psu to the low-latitude box. Shown are the first
1000 years of 4 models (fixed fluxes, #1; fixed T, #2; radiative restoring,
#3, and NSM, #8), represented by the time series of meridional temperature gradient (Fig. 9a), salinity gradient (Fig. 9b), the flow strength (and
hence density gradient, Fig. 9c), and, in Fig. 9d, the combination of T
and S orthogonal to density, denoted 1/ here and called spiciness by Munk
(1981, p. 282; Olbers et al., 1985, used the term veronicity). The models with strong Newtonian relaxation (#2, #8) return to the equilibrium
monotonically. Model #3 shows slight overshoot in all quantities, while
model #1 amplifies the 0.2 psu initial anomaly to excursions of 3.5°C and
0.7 psu. Density gradient and flow rate return to equilibrium monotonically, but spiciness changes sign and reaches quadrupled magnitude of the
original perturbation.
That the flow anomaly in model #1 decays more slowly than in any
other model is plausible from the feedback analysis of section 3: All feedbacks change sign when the low-latitude sinking equilibrium is considered,
so model #1 lacks both negative feedbacks associated with changes in airsea fluxes. The growth in spiciness gradient, however, which by far outweighs the decay in density gradient, is less expected since model #1 has
only this one steady state, with the given set of surface fluxes, and all
eigenvalues are negative (see below). The temporary growth in anomalies
is caused by the interference between the eigensolutions of the non-normal
operator (only normal matrices, i.e., for which AAT = AT A, have orthogonal eigenvectors; see Trefethen et al., 1993, for a general discussion of
non-normality in hydrodynamic stability problems); it is readily shown
that the original perturbation is small enough to ensure linear behavior so
nonlinear effects play no role.
The perturbation solution of model #1 is now given explicitly. Define
density and spiciness gradients and their forcing as
p
aT - {3S
(74)
1/
aT+ {3S
(75)
Hp = aHT - {3Hs
(76)
Hv = aHT +{3Hs
(77)
Notice that the density difference is defined as high-latitude minus lowlatitude, in contrast to all other quantities. Hence,
high-latitude box and -0.2 psu to the low-latitude box. Shown are the first
1000 years of 4 models (fixed fluxes, #1; fixed T, #2; radiative restoring,
#3, and NSM, #8), represented by the time series of meridional temperature gradient (Fig. 9a), salinity gradient (Fig. 9b), the flow strength (and
hence density gradient, Fig. 9c), and, in Fig. 9d, the combination of T
and S orthogonal to density, denoted 1/ here and called spiciness by Munk
(1981, p. 282; Olbers et al., 1985, used the term veronicity). The models with strong Newtonian relaxation (#2, #8) return to the equilibrium
monotonically. Model #3 shows slight overshoot in all quantities, while
model #1 amplifies the 0.2 psu initial anomaly to excursions of 3.5°C and
0.7 psu. Density gradient and flow rate return to equilibrium monotonically, but spiciness changes sign and reaches quadrupled magnitude of the
original perturbation.
That the flow anomaly in model #1 decays more slowly than in any
other model is plausible from the feedback analysis of section 3: All feedbacks change sign when the low-latitude sinking equilibrium is considered,
so model #1 lacks both negative feedbacks associated with changes in airsea fluxes. The growth in spiciness gradient, however, which by far outweighs the decay in density gradient, is less expected since model #1 has
only this one steady state, with the given set of surface fluxes, and all
eigenvalues are negative (see below). The temporary growth in anomalies
is caused by the interference between the eigensolutions of the non-normal
operator (only normal matrices, i.e., for which AAT = AT A, have orthogonal eigenvectors; see Trefethen et al., 1993, for a general discussion of
non-normality in hydrodynamic stability problems); it is readily shown
that the original perturbation is small enough to ensure linear behavior so
nonlinear effects play no role.
The perturbation solution of model #1 is now given explicitly. Define
density and spiciness gradients and their forcing as
p
aT - {3S
(74)
1/
aT+ {3S
(75)
Hp = aHT - {3Hs
(76)
Hv = aHT +{3Hs
(77)
Notice that the density difference is defined as high-latitude minus lowlatitude, in contrast to all other quantities. Hence,