3
Wave, mean ‡ow eqs for QG dynamics
3.1
The quasigeostrophic equations on a -plane
We consider motion in a Boussinesq ‡uid with characteristic horizontal length
scale L, velocity scale U , time scale L=U on a -plane for which the Coriolis
parameter is f = f0 + y. We make the assumptions that:
(i) the Rossby number Ro = U=f0 L is small,
(ii)
L=f0 Ro (which, in a spherical context, means that L << a, where a
is the Earth’s radius),
(iii) the isopycnal slopes j@b=@xj=j@b=@zj and j@b=@yj=j@b=@zj (where b =
are Ro (otherwise vertical motions would not be small), and
g = 0)
(iv) the static stability N 2 = @b=@z is a function of z only [this stems from
(iii)].
Under these assumptions, we …nd that the leading order equations give
geostrophic balance, in which we write the geostrophic velocities [i:e:, the leading
order terms in a Rossby number expansion of (u; v; w)] as
u=
where
@
@
;v=
;w=0;
@y
@x
(1)
is the geostrophic streamfunction:
=
[p
p0 (z)]
.
0 f0
(2)
Hydrostatic balance then gives us
@
1
= b.
@z
f0
(3)
At next order, we obtain our quasigeostrophic equations. The equations of
motion are,
Dg u
yv f0 va = Gx ,
(4)
Dg v + yu + f0 ua = Gy ,
where Dg is the time derivative following the geostrophic ‡ow
Dg
@
@
@
+u
+v
;
@t
@x
@y
(ua ; va ; wa ) is the ageostrophic velocity (i:e:, the di¤erence between the actual
velocity and the geostrophic one), and (Gx ; Gy ) is the external (e.g. frictional,
winb stress) force per unit mass. Similarly, the buoyancy equation is
Dg b + wa N 2 = B
1
(5)
where B is the nonconservatie buoyancy source (such as smallscale mixing).
From these, we can readily derive the equation for quasigeostrophic potential
vorticity, q :
Dg q = X ,
(6)
where
q = f0 + y +
@v
@x
and
X =
and where
@u
@
+ f0
@y
@z
@G
y
@x
b
S
@G
@
x + f0
@y
@z
@2
@2
@
+
+
2
2
@x
@y
@z
2
= f0 + y +
B
S
2
,
f02 @
N 2 @z
(7)
(8)
.
(9)
Eq. (6) tells us that, for conservative ‡ow (G = 0, B = 0, whence X = 0), q is
conserved following the geostrophic ‡ow. When the ‡ow is not conservative, X
represents the local sources or sinks of q arising from viscous and/or diabatic
e¤ects.
3.2
PV ‡uxes and the Eliassen-Palm theorem
Consider small-amplitude motions on a steady, zonally-uniform basic state
U = U (y; z)
B = B(y; z)
= (y; z)
where
@
= U;
@y
@B
@U
= f0
.
@y
@z
The basic state PV is:
Q(y; z) = f0 + y +
@2
@
+
2
@y
@z
2
= f0 + y +
f02 @
N 2 @z
.
(10)
Write
=
where
0
+
0
(x; y; z; t)
0
is a small perturbation. Now,v = @
q0 =
2
0
=
0
=@x and
@2 0
@2 0
@
+
+
2
@x
@y 2
@z
2
f02 @ 0
N 2 @z
(11)
so
@ 0 @2 0
1 @
=
@x @x2
2 @x
"
@ 0
@x
@ 0 @2 0
@ @ 0@ 0
=
@x @y 2
@y @x @y
@ 0@ 0
@x @y
@
@y
#
=0;
@ 0 @2 0
@y @x@y
"
1 @
@ 0
2 @x
@y
@ @ 0@ 0
=
@y @x @y
=
2
2
#
;
and
f02 @ 0 @ 2 0
N 2 @z @x@z
"
f02 @
@ 0
2
2N @x
@z
@ f02 @ 0 @ 0
@ 0 @ f02 @ 0
=
@z N 2 @x @z
@x @z N 2 @z
@ f02 @ 0 @ 0
=
@z N 2 @x @z
=
@ f02 @ 0 @ 0
@z N 2 @x @z
2
#
:
Therefore, from (11),
v0 q0 = r F .
where
F=
Fy
Fz
=
!
@ 0 @ 0
@x @y
2
f0 @ 0 @ 0
N 2 @x @z
(12)
=
u0 v 0
:
f0 0 0
N2 v b
(13)
F is known as the ELIASSEN-PALM ‡ux. Note that the northward component
of F is (minus) the northward ‡ux of zonal momentum by the waves, while the
vertical component is proportional to the northward ‡ux of buoyancy, v 0 b0 :
Now, linearizing the quasigeostrophic potential vorticity equation (6) gives
@
@
+U
@t
@x
q0 + v0
@Q
= X0 ;
@y
multiply by q 0 and average:
@
@t
1 02
q
2
+ v0 q0
@Q
= v0 X 0 :
@y
(14)
If we de…ne
A=
1 02
q =
2
@Q
@y
D = v0 X 0 =
and
3
@Q
@y
;
then
@A
+r F=D .
(15)
@t
Eq. (15) is the ELIASSEN-PALM RELATION. It is a conservation law for
zonally-averaged wave activity whose density is A. Note that D ! 0 for conservative ‡ow.
The signi…cance of this relation is that it gives us a measure of the ‡ux of
wave activity through wave propagation. For example, if the waves are conservative (D = 0) then A must increase with time wherever F is convergent and
decrease wherever it is divergent. Thus F is a meaningful measure of the propagation of wave activity from one place to another. This becomes most obvious
for almost-plane waves (WKB theory) when, as we shall see,
F = cg A
where cg is group velocity, in which case
@A
+ r cg A = D .
@t
However, note that F remains valid as a measure of the ‡ux of wave activity
even when WKB theory is not valid and we cannot even de…ne group velocity.
[N.B. there are some subtleties to these arguments if @Q=@y changes sign
anywhere, since A is then not positive de…nite (and so increasing A does not
necessarily mean increasing wave amplitude). Indeed, if this occurs, the basic
state may be unstable— the Charney-Stern necessary condition for instability
can in fact be readily obtained from (15).]
3.3
The Eliassen-Palm theorem
For waves which are steady (@A=@t = 0), of small amplitude and conservative
(D = 0) , the ‡ux F is nondivergent. This is, through (12), the same thing as
saying that the northward ‡ux of quasigeostrophic potential vorticity vanishes
under these conditions [a result we could of course have obtained from (14)
without involving F]:
3.4
Potential vorticity transport and the nonacceleration
theorem
We now consider the problem of how eddies (in this case, quasigeostrophic
eddies) impact on the zonal mean circulation. We return to eddies which may
not be small and to the QGPV budget, which becomes, on zonal averaging
@q
@ 0 0
+
(v q ) = X .
@t
@y
(16)
Note that, unlike (4.8), there is no mean advection term in (16). This is because
there is no advection by w in this quasigeostrophic case and v = @ =@x = 0.
Similarly, the eddy ‡ux contains no vertical component.
4
The in‡uence of the eddies on mean potential vorticity, therefore, is described
entirely by the northward ‡ux v 0 q 0 (unlike the case of the general conservative
tracer we discussed in Section 4, where the waves could also impact the mean
budget by in‡uencing u). Given the equivalence between v 0 q 0 and r F, we
could make the same statement about r F— which informs us immediately
that F is telling us something about wave transport as well as propagation.
Now, we know from the Eliassen-Palm theorem that if the waves are everywhere
(I) of steady amplitude,
(II) conservative, and
(III) of small amplitude, (so that terms of O( 3 ) in the wave activity budget
are negligible)
then F is nondivergent and v 0 q 0 = 0. Under these conditions, therefore, the
budget equation for zonally-averaged QGPV is independent of the waves (if we
assume that X is so independent). Now,
q=f+
2
( )
therefore
2
@ =@t = @ q=@t = X :
If @ q=@t is independent of the waves, then so is 2 [@ =@t]; since 2 is an
elliptic operator, the solution of the above equation for @ =@t invokes boundary
conditions on @ =@t. If, however, we invoke the further condition that
(IV) the boundary conditions on @ =@t are independent of the waves
then @ =@t is everywhere independent of the waves. Since u = @ =@y and
b = @ =@z , it then follows that, under conditions (I)-(IV), the tendencies of the
mean geostrophic ‡ow, mean buoyancy and mean QGPV are independent of the
waves. This is known as the nonacceleration theorem (or, sometimes, the
“nontransport” or (less accurately) “noninteraction” theorem) and conditions
(I)-(IV) are sometimes known as “nonacceleration conditions”.
This result (which, as we shall see, can be a surprising one under some
circumstances) is suggested via GLM theory, even for …nite amplitude waves,
by Kelvin’s circulation theorem. Consider Fig 3.4; a connected, horizontal,
material curve is disturbed by the appearance of a wave disturbance (in the
absence of that disturbance, the curve would lie along a latitude circle). Now,
the circulation theorem states that
Z
d
u dl = 0
dt C
for conservative ‡ow [condition (II)] (note that there is no solenoidal term in our
pressure coordinates under quasigeostrophic assumptions). That is, even if the
5
curve C is severely distorted by the waves, the circulation around the contour is
una¤ected. In our GLM perspective, this means that
@ uL
=0.
@t
Extension of this to prove our result for the Eulerian average from the GLM
average along C invokes conditions (I) and (III) so that @ uS =@t = 0(cf . the
arguments in Section 4 leading to @ bS =@t = 0 in similar circumstances).
3.5
Mean momentum and buoyancy budgets — conventional approach
In a similar fashion, we can take the zonal mean of the quasigeostrophic momentum and buoyancy equations. The zonal mean of the …rst of (4) gives us
@u
@t
f0 va = G x
@ 0 0
(u v ) ,
@y
(17)
where we have used v = 0 (the mean ageostrophic wind is not zero, however).
Similarly, the buoyancy equation (5) gives
@b
+ wa N 2 = Q
@t
@ 0 0
(v b ) .
@y
(18)
The mean ‡ow equations are closed by the continuity equation
@v a
@wa
+
=0
@y
@z
(19)
and the thermal wind shear equation
f0
@u
=
@z
6
@b
.
@y
(20)
(The two latter equations are linear and so the zonal averaging is trivial).
The evolution of the zonal mean state in the presence of eddies is therefore
speci…ed by (17-20). In this case, the e¤ects of wave transport are manifested
in two terms — the convergence of the eddy ‡uxes of momentum u0 v 0 and
buoyancy. Both these terms force the mean ‡ow equations and it is important
to note that the whole system is coupled, i:e:, the buoyancy ‡uxes can impact
on the mean winds just as much as can the momentum ‡uxes. Thermal wind
balance requires this to be true. Consider, for example, a wave with v 0 b0 6= 0
but u0 v 0 = 0 (the simple upward-propagating planetary wave we discussed in
Section 4; it also seems to be largely true in the ocean). The mean state could not
respond with a changing mean buoyancy only; thermal wind balance demands
a corresponding change in u. From (17) this could only be achieved through an
ageostrophic meridional circulation, which would impact on both the momentum
and buoyancy budgets. Thus, the waves will not only drive @u=@t and @b=@t,
but also va and wa (except in the unlikely case where the eddy forcing terms
conspire not to disturb thermal wind balance).
To put the same statements into mathematics, what (17-20) give us is a set
of 4 equations in the 4 unknowns @u=@t, @b=@t, va and wa in terms of the two
eddy driving terms. In general, when both the eddy ‡ux terms are nonzero,
there is no simple way of saying which forcing achieves what response.
Moreover, the central role of the potential vorticity ‡ux— obvious in the
PV budget— is not at all obvious here. Indeed, we have seen from the mean
potential vorticity budget that under “nonacceleration conditions” @u=@t and
@b=@t must be zero under these conditions. What must (and does) happen
under such circumstances is that the eddies induce ageostrophic mean motions
whose e¤ects in (17) and (18) exactly balance the eddy ‡ux terms. (Which is
reminiscent of the similar example we discussed in Section 4 where the eddy
‡uxes seemed to be telling us that our de…nition of mean circulation was not
the most natural one). All in all, this approach to the mean momentum and
buoyancy budgets is not giving us much insight into what is going on.
3.6
Transformed Eulerian-mean theory
These di¢ culties can be bypassed by what may seem to be a mathematical trick
but is in fact based on (but not the same thing as) GLM theory. The “trick” is
to rede…ne the mean meridional (ageostrophic) circulation.
We return to the mean buoyancy budget (18). This is (apart from the
noncvonservative buoyancy forcing term and the loss of some terms through
the quasigeostrophic assumptions) essentially the same as our Eulerian mean
budget for a conserved quantity. We saw in that case that GLM theory yields
a mean budget equation with no explicit eddy terms. Under quasigeostrophic
assumptions1 , we can arrange the same result within an Eulerian framework.
We begin by noting that, from (19), we may de…ne an ageostrophic mean
1 In fact, one can go through this procedure for the nongeostrophic case also. The main
results remain unchanged, though the analysis is more complicated.
7
streamfunction
a
such that
@ a
,
@z
(va ; wa ) =
@ a
@y
.
(21)
Now, we look for a more revealing way of de…ning “mean circulation”. To
make things as simple as possible, we insist that our modi…ed circulation be
nondivergent also, so we write
@
@
,
@z @y
(v ; w ) =
;
(22)
where the new streamfunction is
=
a
+
c:
If we substitute this into (18), we obtain
@ 0 0
@ c
(v b ) + N 2
.
@y
@y
@b
+ w N2 = B
@t
Noting that N 2 = N 2 (z), it follows that if we make the choice
c
so that
v 0 b0
,
N2
=
— the streamfunction of the so-called residual circulation— is
v 0 b0
,
N2
(23)
@b
+ w N2 = B .
@t
(24)
=
a
+
we obtain the buoyancy equation
We have thus succeeded in deriving a mean buoyancy equation in which
there are no explicit eddy terms; buoyancy is transported solely through the
mean vertical “residual” motion. It might be thought, of course, that the eddy
terms are still there, implicit in w ; but this was also true of wa which, as noted
earlier, is in general in‡uenced by the eddies. What we have done is to rede…ne
this in‡uence so as to put the mean buoyancy budget into its simplest possible
form.
We now need to complete our transformed system of equations. The continuity equation is
@v
@w
+
=0
(25)
@y
@z
[we arranged this by (22)]. The thermal wind equation stays as before, i:e: :
f0
@u
=
@z
8
b:
(26)
The momentum equation is less trivial. We need to replace va using (22); the
result is
@u
f0 v = G x + r F ,
(27)
@t
where F is the Eliassen-Palm ‡ux.
This transformation— which is nothing more than a di¤erent way of writing
the same equations— makes the role of the eddies look rather di¤erent. We now
have, in (24-27), a set of equations for v ; w ; @u=@t and @b=@t in which the
only term representing eddy forcing is a term r F = v 0 q 0 , which appears as an
e¤ective body force (per unit mass) in the mean momentum equation. [We could,
for example, rede…ne Gx to absorb this term]. It is clear, therefore, that under
nonacceleration conditions when r F = 0 and the boundary conditions are
independent of wave-dependent terms, v , w , @u=@t and @b=@t are independent
of the waves.
When nonacceleration conditions are not satis…ed, the transformed equations
o¤er a more transparent approach to the problem simply because the single
term represented by the e¤ective body force r F entirely summarizes the eddy
forcing of the mean state (there being no thermal eddy forcing to confuse the
issue). In fact, this formulation gives us another interpretation of F: as an eddy
‡ux of (negative, i:e:, easterly) momentum which, because of the properties
we have just described, is a more reliable measure of the wave transport of
momentum than u0 v 0 alone. Moreover, we now see that the Eliassen-Palm ‡ux
gives us a uni…ed picture of wave propagation (through its role in wave activity
conservation) and transport (through its interpretation as a momentum ‡ux).
This perspective is conceptually very powerful, as we shall see.
Finally, note again that, while the perspective which regards F as a momentum ‡ux may seem to be the result of mere mathematical juggling, it should
be remembered that the process of taking a “mean” is an arbitrary one— there
is no unique way of doing it. In fact, if we had taken the GLM means of the
basic equations we would naturally have arrived at a set of equations in which
there appears (a) no eddy buoyancy ‡ux and (b) a single wave-transport term
in the mean momentum equation (in fact a term very much like the EP ‡ux
divergence). The transformed Eulerian mean equations simply give us a clearer
picture of what is going on.
In fact, appealing to GLM ideas gives us a simple physical picture, under
simple conditions, of F as a momentum ‡ux. Consider a material, isentropic
surface C which, in the absence of eddy motions, is purely horizontal; see Fig
3.6. This surface becomes disturbed under the in‡uence of adiabatic, steady,
9
small-amplitude waves. Because the motions are adiabatic, we may choose C
to be a surface of constant b. Now, because the surface is a steady material
one, there is no mass motion across it and therefore no advective transport
across it. There is, however, a stress across it from pressure forces if C is wavy;
locally, there is a force p per unit area acting normally to the surface. If the
surface is inclined (in the x z plane) at an angle (in geometric space) to the
horizontal, then the (x; z) components of this force per unit horizontal area are
p(cos ; sin ). Therefore, the zonally-averaged zonal stress on C is where
0
=
p sin
'
p '
p
(since = 0) where p is the pressure variation along C . But, since is small,
@b @b
@b0
2
tan
@( zg )= x
@x = @z =
@x =N (recall that the surface is one of
constant b); therefore zg , the geometric height variation along C, is given by
zg
b0 =N 2 :
Now, if C is the surface of constant geometric height that is the reference position
for C , and p0 the pressure variation along C, then, along C , p = p0 g 0 zg .
Therefore
@ ( zg ) 0
@ ( zg )
@ ( zg ) 0
p
g 0
zg =
p
@x
@x
@x
@p0
f
zg
= 2 0 v 0 b0 ;
=
@x
N
p=
and therefore
f 0 0
vb .
N2
Thus we see that (in this case, the vertical component of) the Eliassen-Palm ‡ux
has a real physical meaning as a (negative) momentum ‡ux; the waves, through
distortion of material surfaces, exert a stress— a radiation stress— on the ‡uid
above which is just equal to Fz . [If we had chosen a surface in the (x; z) plane,
we would have found the stress on that to be u0 v 0 ]. So the appearance of F as
the eddy momentum ‡ux in the transformed Eulerian-mean formulation is not
just a result of mathematical trickery, but reveals an underlying Lagrangian-like
interpretation of the formulation.
This, in fact, illustrates a crucial di¤erence in the way momentum is transported, compared with conserved tracers. Consider Fig. 3.6. Suppose that,
initially, a set of material lines is undisturbed (dashed lines). A wave is generated at y = y0 , which then propagates to large y; the material lines are then
as marked by the solid curves. Now, because the air that was originally at y0
has moved no further than a distance y, transport of all conserved quantities—
chemical tracers, buoyancy, and potential vorticity— is similarly limited to a
distance y: these quantities cannot be transported into the far …eld. However,
since the wave has propagated to the far …eld, and since the consequent waviness of the material lines will, in general, be associated with a radiation stress,
momentum can be transported into the far …eld. In other words, conserved
quantities can only be advected; momentum can be radiated.
=
10
3.7
Example: our simple, vertically propagating Rossby
wave
In the simple example of Section 2.3, we had a steady, small-amplitude vertically
propagating Rossby wave in the absence of friction or buoyancy sources and
sinks. So, provided the boundary conditions on the transformed problem are
independent of wave terms (we shall see that they are), the nonacceleration
theorem must be satis…ed, and there is no tendency to accelerate the mean
‡ow, nor to change the mean buoyancy structure.
For this case, we had the O(") wave solution
0
where 6
= Re
ei(kx+mz) sin ly
is a constant. Then
u0 =
@ 0
=
@y
Re l
ei(kx+mz) cos ly
@ 0
= Re ik ei(kx+mz) sin ly ;
@x
@
@ 0
w0 = U N 2
f
= f kmU N
@x
@z
v0 =
Since b0 = f @
0
2
Re
ei(kx+mz) sin ly :
=@z,
b0 = f m Re i
ei(kx+mz) sin ly :
Therefore
1
1
Re ( l cos ly) (ik sin ly) =
kl sin 2ly 2 Re [i] = 0 ;
2
4
1
1
v 0 b0 = Re (ik sin ly) (if m sin ly) = f mk 2 sin2 ly :
2
2
u0 v 0 =
11
Hence the EP ‡ux is
F=
Fy
Fz
=
u0 v 0
0
=
f 0 0
N2 v b
f 2 mk
2N 2
2
sin2 ly
:
So the y-component is zero: this …ts with the concept of F as a measure of
wave propagation, since our channel geometry (and the construction of this
example as a single cross-channel mode) inhibits latitudinal propagation. The
z-component is nonzero, and is positive: the wave is propagating upward. [N.B.
Now we can see that the wave must possess a poleward buoyancy ‡ux v 0 b0 if itis
upward propagating, whether or not there is any poleward gradient of b.] Note,
however, that Fz is independent of z: therefore r F = 0, in accord with the
prediction of the Eliassen-Palm theorem. Therefore there is no forcing of the
TEM momentum eq. (27), nor in any of the other eqs. in our transformed set.
But what of the boundary conditions? On the side boundaries, we have
v = 0, whence v = 0. But v = v @ v 0 b0 =N 2 =@z = 0 also, since v 0 = 0. At
the top, we can impose boundedness for both w and w . The bottom is more
problematic. It is tempting to assume that w = 0 there, since there can be no
‡ow through the boundary, but, since
wjz=0 =
we have
wjz=0 = u0
Since b0 =
dh
,
dt
@h0
@h0
@
+ v0
=
v 0 h0 :
@x
@y
@y
N 2 h0 on the boundary, we have
@
@y
wjz=0 =
v 0 b0
N2
:
(28)
So the mean ‡ow is nonzero on the boundary! It subsides into the boundary
at low latitudes, and upwells out of the boundary at high latitude— why? Consider Fig. ??. Above the level of the highest topography, the mean v is just
the integral around a complete latitude circle, so v
O "2 , since the O(")
wave component integrates to zero. But, below, the mountain tops, there is a
poleward O(") ‡ow (since v 0 b0 > 0, so v 0 and h0 are out of phase) which does
not integrate to zero, because there is no compensating equatorward ‡ow below
ground: there is a net poleward ‡ow down the valleys. This ‡ow is O("), its
depth is O("), so the net volume ‡ux is O("2 ), and in fact it is (of course) exactly
what is required to balance the mean subsidence into/out of the boundary [it
is not hard to show this]. Note that, since h0 is in…nitessimally small under our
assumptions, all this appears to happen in a delta-function on the boundary at
z = 0.
Now consider the residual ‡ow. We have, from (23)
w =w+
@
@y
12
v 0 b0
N2
so, on z = 0, from (28)
w jz=0 =
@
@y
v 0 b0
N2
@
@y
v 0 b0
N2
=0
so the residual circulation vanishes at the boundary. [This is because our boundary is one of constant b— as we are about to see, we do not get this result if
b is nonzero along the boundary.] So the solution to our full set of TEM eqs.,
consistent with the boundary conditions, is
@u
@b
=
=v =w =0:
@t
@t
The wave thus has no impact on the transformed mean state.
From the conventional mean eqs, the buoyancy budget is given by (18), and
is
@b
@ 0 0
+ wa N 2 =
(v b ) .
@t
@y
13
But we have seen that v 0 b0 is nonzero, and yet @ b=@t = 0. So we must have a
mean ageostrophic meridional circulation
wa =
@ v 0 b0
(
)
@y N 2
whish is of course consistent with wa = w
@ v 0 b0 =N 2 =@y. So the waves do
have an impact on the mean state (in the conventional view), despite nonacceleration2 . The mean ageostrophic circulation, as we have seen, is downward in
low latitudes and upward in high latitudes: this is a simple version of the Ferrel
cells in the atmosphere or the Deacon cell in the Southern Ocean.
3.8
Boundary conditions on a non-isentropic surface (where
b0 6= 0)
By contrast with our topographically-forced case, if the lower boundary is ‡at
but there is a nonzero buoyancy ‡ux there, then
wjz=0 = 0 ;
(since w = 0 everywhere on the boundary, and the boundary is ‡at, so there
are no surprises associated with the zonal averaging) but
w jz=0 = wjz=0 +
@
@y
v 0 b0
N2
=
@
@y
v 0 b0
N2
is nonzero there: if the buoyancy ‡ux is poleward, there is, e.g., a mean residual
‡ow into a lower boundary in low latitudes, and out of it at high latitudes.
There is an implied equatorward ‡ow within a delta-function at the boundary
(within Bretherton’s PV sheet); we’ll understand what this ‡ow is when we
consider the problem in entropy/density coordinates. Basically, we have in the
two cases:
1. Lower boundary ‡at in entropy (density) coordinates ! w = 0 ; w 6= 0
2. Lower boundary ‡at in geometric (z) coordinates ! w 6= 0 ; w = 0.
3.9
The EP ‡ux and group velocity
We deduced from (15) that F is a measure of wave propagation, and noted that
it has a relationship to group velocity. In our QG case, Rossby waves are the
waves of interest3 . Group velocity can only be de…ned for almost-plane waves,
so consider the plane wave solution
0
= Re
ei(kx+ly+mz
!t)
2 This is why it is not really accurate to refer to the nonacceleration theorem as “noninteraction”, as some people do.
3 The result to follow is true, in the general case, for other classes of waves, too.
14
where ! satis…es the Rossby wave dispersion relation (assuming constant mean
zonal ‡ow U and @ q=@y = )
k
! = kU
2
q
f2
2 is a modi…ed total wavenumber. Note that the
where = k 2 + l2 + N
2m
meridional components of group velocity are
@! @!
;
@l @m
cg =
k
=2
l;
4
f2
m
N2
:
Now, since
@2
f 2 @2
@2
+ 2+ 2 2
2
@x
@y
N @z
q0 =
0
the wave activity density is
A=
4
q 02
=
2
4
2
:
The EP ‡ux components are
Fy =
=
u0 v 0 =
1
kl
2
2
@ 0@ 0
@x @y
;
and
f 0 0
f @ 0@ 0
vb = 2
2
N
N @x @z
f2
km 2 :
=
2N 2
Fz =
Therefore
F = (Fy ; Fz ) =
k
2
2
4
= cg
l;
f2
m
N2
2
4
= cg A :
Thus, the ‡ux F indeed corresponds in this case to the wave activity, with
density A, moving with velocity cg .
15
3.10
3.10.1
The EP ‡ux as a momentum ‡ux, and momentum
conservation
Pseudomomentum
From the mean momentum equation (27),
@u
@t
f0 v = G x + r F ;
the EP ‡ux acts as an e¤ective ‡ux of (negative) momentum, and so it is
already apparent to us that the eddies act to transport mean momentum in
the ‡uid (and we have seen how we can interpret that in terms of form drag on
isopycnal/isentropic surfaces). In fact, if we susbstitute for r F from the wave
activity conservation eq. (15), we have
@ (u + A)
@t
f0 v = G x + D :
(29)
So, in a nonrotating ‡uid (or if we ignore for now the meridional circulation
which, as we’ll see later, basically acts to redistribute momentum locally) then
for conservative ‡ow, u + A is constant. If a wave, initially absent (A = 0),
appears in a certain region, then it will induce a velocity change u =
A. So
the wave behaves as if4 it carries a momentum per unit mass A. In generalized
Lagrangian mean theory (Andrews & McIntyre, JFM, 1978) the quantity equivalent to A is known as the pseudomomentum density, and its conservation is
related to the invariance of the mean ‡ow to translation in the x-direction, just
as conservation of momentum in general is related to the translational invariance of the whole problem5 . ( A is sometimes referred to in the literature as
pseudomomentum, but it is not strictly the same thing.)
Note that (29) implies that (again, neglecting the Coriolis term, for now)
if both the mean state and waves are conservative, any induced mean ‡ow
changes are temporary, or reversible, in the sense that any change induced by
the appearance of the wave disappears when the wave goes away. Irreversible
changes in the mean state require nonconservative e¤ects.
3.10.2
Momentum conservation
If we integrate (27) over the whole domain R [z1 < z < z2 ; y1 < y < y2 ]
bounded by boundary C, then
ZZ
I
Z y2
ZZ
@
u dy dz =
F n dl +f0
[ (y; z2 )
(y; z1 )] dy +
G x dy dz ;
@t
R
C
y1
R
4 Of course, in an incompressible ‡uid, the wave does not actually have any momentum,
since by de…nition u0 = 0.
5 If we took a time-, rather than zonal-, average, the corresponding conservation is of
pseudoenergy. This is less useful than pseudomomentum, as it is not sign de…nite in realistic
circumstances, as pseudomomentum often is.
16
where n is the outward unit normal at the boundary. The integrated mean momentum of the whole system can change only through a net in‡ux of momentum
by the eddies (via F), by advection of angular momentum into the system by
the residual circulation, and by frictional or other body forces.
The momentum balance helps us understand the boundary issues we discussed earlier. In our mountain-forced Rossby wave case, there is a nonzero F
at the bottom boundary which corresponds to an injection of (negative) momentum into the system. This is no problem— the implied momentum sink
corresponds exactly to the form drag on the topography. But for a ‡at boundary with nonzero heat ‡uxes, and hence also nonzero F, there is also an implied
net ‡ux which does not correspond to a form drag. So there has to be an
exactly opposite ‡ux through the residual circulation— which must therefore be
nonzero on the boundary.
3.11
3.11.1
The Charney-Stern stability theorem in terms of wave
activity conservation
Stability
Consider now the globally-integrated budget of our psuedomomentum-like wave
activity. From (15),
ZZ
I
ZZ
@
A dy dz + F n dl =
D dy dz :
@t
R
C
R
For conservative ‡ow (D = 0) integrated wave activity can change only if there
is a ‡ux of wave activity through the boundaries. If these ‡uxes are zero, which
requires
1. rigid sidewalls: v = 0 ! Fy =
u0 v 0 = 0, and
2. upper and lower boundaries of uniform density (entropy): b0 = 0 ! Fz =
f v 0 b0 =N 2 = 0,
then
@
@t
ZZ
A dy dz = 0 :
R
So net wave activity cannot then grow or decay— it can merely be redistributed.
@q
, if the PV gradient is single-signed and then we
Now, since A = 12 q 02 = @y
obviously have a meaningful constraint on how a disturbance can grow. In
fact, if we look for normal mode growth such that q 02 = S(y; z)T (t) (where we
may de…ne both S and T to be positive de…nite, then
ZZ
dT
S
dy dz = 0 :
dt
R @ q=@y
Then, if the PV gradient is single-signed, the integral cannot vanish, so dT =dt =
0: the mean state is stable to normal mode disturbances. So we can now
17
understand this result in terms of wave activity conservation. (In fact, the
derivation is essentially the same as the Charney-Stern derivation: they did not
describe it in terms of wave activity, as the language was not available at that
time.) We can also understand the requirement for isopycnal/isentropic upper
and lower boundaries as a requirement of no wave activity ‡ux through the
boundaries.
While this constraint does not apply to non-normal-mode growth, note that
wave activity consrvation places severe constraints on the growth of disturbances
if the PV gradient does not change sign anywhere.
3.11.2
Instability
Under most atmospheric and oceanic conditions, @ q=@y > 0 and instability
arises from localized regions where the gradient becomes negative. Since, for
conservative ‡ow, r F = @A=@t, for a growing disturbance F is convergent
where @ q=@y > 0 and divergent where @ q=@y < 0. So, e.g., if @ q=@y < 0 in a
localized region indicated by the grey shading in Fig. 1, the ‡ux will be divergent
z
y
Figure 1: Schematic of F.
there and will look qualitatively as shown in the …gure. [If the instability is
primarily barotropic, the ‡uxes will be dominated by the horizontal component
(i.e. that corresponding to u0 v 0 ); if baroclinic, by the vertical component (corr.
to v 0 b0 ):]
In the Eady problem, @ q=@y = 0 everywhere and the instability arises
through the boundary terms. Clearly (since r F =v 0 q 0 = 0) F is nondivergent everywhere! The ‡ux is nonzero, however, because it appears out of the
bottom boundary and disappears into the top boundary, as shown in Fig. 2(a).
We can see this in one of 2 ways. But …rst, note that u0 v 0 = 0 for the classical
Eady solution (no N-S asymmetry), so we only need think about the vertical
component.
1. Since we know that the buoyancy ‡ux is poleward, we have Fz = Nf2 v 0 b0 >
0 everywhere. (In fact, since r F = @Fz =@z = 0, Fz is constant with z).
So the wave activity just appears out of the bottom boundary, propagates
upward, and disappears at the top, as shown. Note that the vertical
18
Figure 2: (From Edmon et al., JAS, 1980.)
19
residual velocity is also nonzero at the boundaries and, in fact, vertical
everywhere.
2. If we take Bretherton’s view of the boundaries as isopycnal surfaces with
“PV sheets” immediately inside them where @ q=@y is nonzero (negative
on the bottom, positive on the top) then F is zero on the boundaries,
but divergent within the bottom PV sheet and convergent within the top
sheet. (The return ‡ow of the residual circulation takes place within these
sheets.)
The Charney problem (@ q=@y > 0 in the interior, < 0 within a PV sheet on
the bottom boundary) is intermediate (Fig. 2(b)), with F appearing out of the
bottom boundary and convergent everywhere in the interior.
3.12
Observed EP ‡uxes and residual circulation in the
troposphere
The EP ‡ux for a compressible gas in spherical geometry can be written as
F=
F
Fz
=
2
cos u0 v 0
sin cos v 0 0 =@ @z
in log-pressure coordinates, where here z = H ln (p=1000hPa) (where H is a
constant) and is potential temperature. The factor cos arises because F now
corresponds to a ‡ux of negative angular momentum (strictly, aF does so, where
a is the Earth radius). The density factor is important: the QG momentum eq
is now
@u
1
2 v sin = r F + Gx :
@t
Examples of F and r F for transient eddies from two (old) climatologies
are shown in Fig. 3, for the northern hemisphere in winter and in summer. F
is directed primarily upward, as expected, corresponding to a ‡ux of wave activity from at or near the bottom boundary to the upper troposphere, much as
expected from the Eady or Charney models. The key di¤erence from the simple
models is that, in the real world, the EP ‡uxes refract equatorward in the upper
troposphere. (They are slightly poleward at high latitude, especially in summer.) In the absence of the lateral boundaries of the models, one would expect
the wave activity to spread out laterally from the source region. In addition
to this is the spherical geometry e¤ect whereby waves tend to be refracted toward the equator. (This means that the “eddy momentum ‡ux”u0 v 0 is directed
poleward in the upper troposphere.) This tendency for equatorward refraction
is illustrated in Figs. 4 and 5 for a weakly damped, stationary Rossby wave
propagating from a localized source at (lat,long) = (30; 90), for two cases: solid
body rotation (wave propagates through tropics) and with a realistic upper tropospheric wind pro…le (wave cannot propagate through tropical easterlies, gets
damped in northern tropics).
20
Figure 3: From Edmon et al. (JAS, 1980).
21
Figure 4: Wave propagation from a localized source with zonal winds in solid
body rotation. [I. M. Held]
Figure 5: Wave propagation from a localized sourse with realsitic (upper tropospheric) zonal winds. [I. M. Held]
22
Figure 6: From Edmon et al. (JAS, 1980).
23
The observed residual circulation, for the same climatologies as Fig. 3, is
shown in Fig. 6. The Ferrel cell of the conventional mean has (more or less)
vanished: the tropical Hadley cell still dominates, but now there is a second,
thermally direct, cell in the extratropics. Note, in accordance with our earlier
comments about the lower boundary condition, the residual circulation ‡ows
into and out of the boundary.
24