346
3 Stability of the high-latitude sinking equilibrium
3.1 Atmospheric transport anomalies
The MS model with linear atmospheric transports is a convenient tool to
define a reasonable stable equilibrium of the coupled box model. We now
return to the general power laws for meridional atmospheric transports,
eqs. (9) and (10). Following the strategy already used by NSM, we choose
the coefficients, Xn and i'm, such that models with arbitrary m and n have
the high-latitude sinking equilibrium identical to the model with m = n =
1:
if Xn == xI'i,l-n
if i'n == i'lT 1 - m ,
(33)
(34)
meaning that for a given steady-state temperature gradient T, the meridional transports are identical for arbitrary powers nand m. All other coefficients are unchanged. In particular, the longwave loss does not change
between different models. By construction, then, all models have the same
steady-state temperature and the same steady-state surface fluxes, hence
also the salinities and the flow strength are identical. This equilibrium
common to all models is an ideal starting point for investigating the effect
of the atmospheric transports on the transient behavior.
Notice that a model with arbitrary nand m could have been chosen
to define the reference steady state. In general, it is required that the
relationship
(35)
hold (similarly for moisture transports); eq. (33) is the special case n2 = 1.
Notice also that while all models with coefficients defined by eqs. (33) and
(34) share the high-latitude sinking equilibrium, there is no guarantee that
the steady state is stable to infinitesimal perturbations for all n or m.
Numerical solutions for various nonlinear atmospheric transports will be
presented below. For analytical considerations it is convenient to linearise
the atmospheric transport anomalies about the equilibrium temperature
gradient T,
3 Stability of the high-latitude sinking equilibrium
3.1 Atmospheric transport anomalies
The MS model with linear atmospheric transports is a convenient tool to
define a reasonable stable equilibrium of the coupled box model. We now
return to the general power laws for meridional atmospheric transports,
eqs. (9) and (10). Following the strategy already used by NSM, we choose
the coefficients, Xn and i'm, such that models with arbitrary m and n have
the high-latitude sinking equilibrium identical to the model with m = n =
1:
if Xn == xI'i,l-n
if i'n == i'lT 1 - m ,
(33)
(34)
meaning that for a given steady-state temperature gradient T, the meridional transports are identical for arbitrary powers nand m. All other coefficients are unchanged. In particular, the longwave loss does not change
between different models. By construction, then, all models have the same
steady-state temperature and the same steady-state surface fluxes, hence
also the salinities and the flow strength are identical. This equilibrium
common to all models is an ideal starting point for investigating the effect
of the atmospheric transports on the transient behavior.
Notice that a model with arbitrary nand m could have been chosen
to define the reference steady state. In general, it is required that the
relationship
(35)
hold (similarly for moisture transports); eq. (33) is the special case n2 = 1.
Notice also that while all models with coefficients defined by eqs. (33) and
(34) share the high-latitude sinking equilibrium, there is no guarantee that
the steady state is stable to infinitesimal perturbations for all n or m.
Numerical solutions for various nonlinear atmospheric transports will be
presented below. For analytical considerations it is convenient to linearise
the atmospheric transport anomalies about the equilibrium temperature
gradient T,