The Quasi-Geostrophic Model
101
that W* is continuous across Zn and could be equally well evaluated on either
side of the interface. Thus we could write (3.2.18) equally well as:
W * (zn) = Wn-1 (zn) - ~; - Un-1 · 'Vzn,
or
(3.2.19)
The velocity W* (zn) is positive whenever fluid from layer n crosses the
interface into layer n- 1. This does not require that the vertical velocity, Wn,
itself be positive if there is strong motion across the interface. If fluid crosses
the interface between the layers, a sudden density transformation must take
place at the interface as fluid is changed, say at z = Zn, from a density Pn to a
density of Pn-I· This implies that nonadiabatic processes must be active at the
interface if W* -1- 0. If W* (zn) is positive, this does not, however, imply that the
vertical velocity at the interface Wn (zn) must be positive, only that some process
allows fluid to transform its density nonadiabatically at the interface. Figure
3.2.1 shows three helpful examples. In the first, the interface is fiat and
motionless. The vertical velocity in the fluid is positive, and fluid crosses the
interface, changing its density as it does so. In this case the vertical velocity and
the cross-interface velocity are both positive and in fact equal to each other. In
Fig. 3.2.1 b, the fluid is absolutely stationary, and the interface descends. The
vertical velocity is zero, but the cross-interface velocity is again positive as the
upper layer absorbs lower layer fluid whose density changes in the process. In
the last example (Fig. 3.2.1c) the interface slopes downwards to the right more
steeply than the three-dimensional velocity vector. In this case the vertical
velocity is negative, and yet the cross-interface velocity is again positive.
Fig. 3.2.1a-{:. Schematic presentation of the relationship between the vertical velocity and the crossinterface velocity W* (zn) in three extreme cases. a
The interface is flat and stationary. The crossisopycnal velocity equals the vertical velocity. b The
fluid is stationary, and the interface descends. The
vertical velocity is zero and the cross-interface
velocity is positive. c The interface slopes downward
to the right more steeply than the three-dimensional
velocity in the nth layer. The cross-interface velocity
is positive and the vertical velocity is negative
(a) ---+t _____ zn
lwn
--1--------r----~~ ~!)t.t>
w.(Zn)
(c)
101
that W* is continuous across Zn and could be equally well evaluated on either
side of the interface. Thus we could write (3.2.18) equally well as:
W * (zn) = Wn-1 (zn) - ~; - Un-1 · 'Vzn,
or
(3.2.19)
The velocity W* (zn) is positive whenever fluid from layer n crosses the
interface into layer n- 1. This does not require that the vertical velocity, Wn,
itself be positive if there is strong motion across the interface. If fluid crosses
the interface between the layers, a sudden density transformation must take
place at the interface as fluid is changed, say at z = Zn, from a density Pn to a
density of Pn-I· This implies that nonadiabatic processes must be active at the
interface if W* -1- 0. If W* (zn) is positive, this does not, however, imply that the
vertical velocity at the interface Wn (zn) must be positive, only that some process
allows fluid to transform its density nonadiabatically at the interface. Figure
3.2.1 shows three helpful examples. In the first, the interface is fiat and
motionless. The vertical velocity in the fluid is positive, and fluid crosses the
interface, changing its density as it does so. In this case the vertical velocity and
the cross-interface velocity are both positive and in fact equal to each other. In
Fig. 3.2.1 b, the fluid is absolutely stationary, and the interface descends. The
vertical velocity is zero, but the cross-interface velocity is again positive as the
upper layer absorbs lower layer fluid whose density changes in the process. In
the last example (Fig. 3.2.1c) the interface slopes downwards to the right more
steeply than the three-dimensional velocity vector. In this case the vertical
velocity is negative, and yet the cross-interface velocity is again positive.
Fig. 3.2.1a-{:. Schematic presentation of the relationship between the vertical velocity and the crossinterface velocity W* (zn) in three extreme cases. a
The interface is flat and stationary. The crossisopycnal velocity equals the vertical velocity. b The
fluid is stationary, and the interface descends. The
vertical velocity is zero and the cross-interface
velocity is positive. c The interface slopes downward
to the right more steeply than the three-dimensional
velocity in the nth layer. The cross-interface velocity
is positive and the vertical velocity is negative
(a) ---+t _____ zn
lwn
--1--------r----~~ ~!)t.t>
w.(Zn)
(c)