Quasi-Geostrophic Model with Continuous Stratification
147
gfo a P
2
a ( !5 atjl)
q = '+ f3y- - - - = V' t/1 +- - - + f3y.
Po azN 2
az N 2 az
(3.10.6)
The Laplacian in (3.10.6) is the "horizontal" Laplacian a 2 I ax 2 + a 2 I ay 2 and
the Jacobian in (3.10.5) is the Jacobian operator with respect to x andy.
The vertical velocity is given at the upper boundary, z = 0, by the Ekman
pumping velocity so that with the aid of the density equation and the
geostrophic and hydrostatic approximations this boundary condition can be
written as:
(3.10.7)
In this chapter we consider only adiabatic flows for which Q is zero. Our
discussion closely follows that of Young and Rhines (1982).
It is illuminating and convenient to introduce nondimensional variables for
the continuous problem. This allows us to write the circulation problem in a
neater, more compact form, but more importantly it reveals the characteristic
scales of the motion from scaling considerations.
We introduce nondimensional variables temporarily denoted by primes in
the following manner:
t/1= ULt/1'
(x, y) = L(x', y')
z=dz'
w=Ww'
q = f3Lq'
(3.10.8)
where the vertical scale, d, must be determined. The horizontal scale L is
imposed by the horizontal scale of the wind forcing, and the scale for the vertical
velocity, W, is determined by the magnitude of the Ekman pumping.
The Sverdrup relation f3v = f 8w I az, determines the relation between the
scales for the horizontal, U, and vertical velocity, i.e:
f3Ud
W=-.
fo
(3.10.9)
In the expression for the potential vorticity, q (3.10.6), the relative vorticity
(the first term) is small with respect to the contribution from stretching of
planetary vortex lines (the second term), since their ratio can be estimated as
N2d2 I fo2L2 « 1.
On the other hand, in order for there to be motion below the surface the
stretching term must be of the same order as the planetary vorticity term so
that the q isolines can distort enough to form internal pools of geostrophic
contours which are detached from the eastern boundary. Thus, we require that:
147
gfo a P
2
a ( !5 atjl)
q = '+ f3y- - - - = V' t/1 +- - - + f3y.
Po azN 2
az N 2 az
(3.10.6)
The Laplacian in (3.10.6) is the "horizontal" Laplacian a 2 I ax 2 + a 2 I ay 2 and
the Jacobian in (3.10.5) is the Jacobian operator with respect to x andy.
The vertical velocity is given at the upper boundary, z = 0, by the Ekman
pumping velocity so that with the aid of the density equation and the
geostrophic and hydrostatic approximations this boundary condition can be
written as:
(3.10.7)
In this chapter we consider only adiabatic flows for which Q is zero. Our
discussion closely follows that of Young and Rhines (1982).
It is illuminating and convenient to introduce nondimensional variables for
the continuous problem. This allows us to write the circulation problem in a
neater, more compact form, but more importantly it reveals the characteristic
scales of the motion from scaling considerations.
We introduce nondimensional variables temporarily denoted by primes in
the following manner:
t/1= ULt/1'
(x, y) = L(x', y')
z=dz'
w=Ww'
q = f3Lq'
(3.10.8)
where the vertical scale, d, must be determined. The horizontal scale L is
imposed by the horizontal scale of the wind forcing, and the scale for the vertical
velocity, W, is determined by the magnitude of the Ekman pumping.
The Sverdrup relation f3v = f 8w I az, determines the relation between the
scales for the horizontal, U, and vertical velocity, i.e:
f3Ud
W=-.
fo
(3.10.9)
In the expression for the potential vorticity, q (3.10.6), the relative vorticity
(the first term) is small with respect to the contribution from stretching of
planetary vortex lines (the second term), since their ratio can be estimated as
N2d2 I fo2L2 « 1.
On the other hand, in order for there to be motion below the surface the
stretching term must be of the same order as the planetary vorticity term so
that the q isolines can distort enough to form internal pools of geostrophic
contours which are detached from the eastern boundary. Thus, we require that: