Download 2: Transport of heat, momentum, and PV

2
2.1
Transport of heat, momentum and potential vorticity
Conventional mean momentum equation
We’ll write the inviscid equation of zonal motion (we’ll here be using log-pressure coordinates
with z = H ln p=p0 , with H constant) as
du
dt
fv =
@
@x
(1)
and the continuity equation
r
(2)
u=0
where (z) = p=gH. We introduce the “zonal mean”of any variable a as
1
a(y; z; t) =
L
Z
L
a(x; y; z; t) dx ;
(3)
0
(where we assume periodic behavior in x such that a(L; y; z; t) = a(0; y; z; t) — note that
this means that (@a=@x) = 0) and then the eddy component is
a0 (x; y; z; t) = a(x; y; z; t)
a(y; z; t) :
It follows that, by de…nition,
a0 = 0 :
(4)
Note that the de…nition of the “eddy” variable is contingent on the de…nition of “mean.”
Then the mean of (1), invoking (2), becomes
@u
@u
@u
+v
+w
@t
@y
@z
fv =
1
r
u0 u 0 ;
(5)
thus introducing the concept of “eddy momentum ‡ux” u0 u0 = u0 v 0 ; u0 w0 (it has no
zonal component in our zonal mean formulation). The way in which the eddies in‡uence
the mean state is evidently equivalent to a body force r
u0 u0 acting on the mean zonal
motion. This ‡ux term clearly represents the advection, by the eddy component of the
meridional velocity, of the eddy component of zonal momentum. The zonal momentum per
unit mass is u; so the advective ‡ux of momentum across a coordinate surface (of constant
y and z) is uu, whence its zonal mean is uu = uu + u0 u0 ; hence it is natural to regard
u0 u0 as the eddy ‡ux of momentum. It is important to realise, however, that this de…nition
of eddy momentum ‡ux is a consequence of our de…nition (3) of what is meant by “mean,”
speci…cally, that the average is taken at at constant pressure (constant z in log-p coordinates).
1
One could easily de…ne other means: since transport is fundamentally a Lagrangian process,
one might try taking some kind of Lagragian mean, but we will not do that here1 . One can
take a di¤erent integration path in (3) — the way we have done it is to integrate at constant
latitude and height (i.e.; constant pressure) — by using a di¤erent coordinate system, in
which case the whole problem can look rather di¤erent, as we are about to see.
2.2
Wave, mean ‡ow interaction in isentropic coordinates
In (x; y; ) coordinates, the inviscid zonal equation of motion is2
@u
@u
@u _ @u
+u
+v
+
@t
@x
@y
@
fv =
@M
@x
(6)
_ =0
(7)
and that of continuity
@
@
@
@
+
( u) +
( v) +
@t
@x
@y
@
where
=
g 1 @p=@ is the isentropic coordinate density, and
M = cp T + gz =
+
(8)
is the Montgomery potential, with (p) = cp (p=p0 ) the Exner function and = gz. (Note
that z is “real”height here, not log-pressure height.) It follows from (8), with some juggling
and the assumption of hydrostatic balance, that
@M
=
@
(9)
:
Now our de…nition of zonal mean is
1
a(y; ; t) =
L
Z
L
a(x; y; ; t) dx
0
where the integral is now, of course, taken at constant , and the eddy term is
a0 (x; y; ; t) = a(x; y; ; t)
1
a(y; ; t) :
If you are interested in the Lagrangian perspective, there are some additional notes on this on the class
web page. The conceptually powerful “generalized Lagrangian mean”theory of wave, mean ‡ow, interaction
can be found in Andrews & McIntyre, J. Fluid Mech., 1978 (though it is not for the faint hearted!).
2
The derivation of the -coordinate equations is discussed in Appendix 3A in Andrews, Holton and Leovy
[1987] (at the end of Chapter 3).
2
It makes sense to de…ne mean meridional velocities in a density-weighted sense:
1
v =
( v) ; _ =
_
(10)
since then the mean continuity equation has no eddy ‡ux terms:
@
@
@
+
( v )+
@t
@y
@
_
=0:
The momentum equation (6), on multiplication by , use of (7), and zonal averaging, gives
@
@
@
( v u) +
( u) +
@y
@
@t
_ u
@
( v)0 u0
@y
f v =
@
@
_
0
u0
@M
: (11)
@x
The …rst two terms on the rhs are easily identifable as advection by the eddy mass ‡uxes,
but what about the third term? Since @M=@x = 0, it is an eddy term
@M
=
@x
0
@M 0
@x
and, in fact, the term can be written, using the de…nition of
@M
@x
But @M=@ =
1 @p @M
g @ @x
@ 1 @M
=
p
@
g @x
and (8)
=
1 @2M
p
:
g @x@
(p), so
p
@2M
@x@
@
d @p
(p) = p
@x
dp @x
Z
@
d
p
dp = 0 ;
=
@x
dp
= p
hence
@M
@
=
@x
@
1 @M
p
g @x
=
@
@
1 0 @M 0
p
g @x
:
In total, therefore, (11) becomes
@
@
@
( u)+ ( v u)+
@t
@y
@
_ u
f v =
i @
@ h
0 0
( v) u
@y
@
3
_
0
u0
1 0 @M 0
p
g @x
; (12)
there is an eddy momentum ‡ux whose y and
( v)0 u0 ;
_
0
components are given by
1 0 @M 0
p
g @x
u0
:
(13)
The additional, non-advective, term in (13) represents the form stress acting on the
isentropic surface. If one has, e.g., surface topography z(x; y) there is an inviscid stress on
is the angle of the local surface
the earth’s surface given by psin = p@z=@x, where3
with respect to the horizontal; accordingly there is a form drag of p@z=@x acting on the
atmosphere above. Now,
p0
@M 0
@M
d @p
@z
= p
=p
+g
@x
@x
dp @x
@x
d @p
@z
@z
= p
+ gp
= gp
;
dp @x
@x
@x
since we just established that the …rst term is zero. Hence
1 0 @M 0
p
=
g @x
p
@z
:
@x
Since @z=@x here is just the slope of the isentropic surface, we can therefore immediately
identify this extra term as the form drag acting on an isentropic surface due to the presence
of the eddies. So, in a di¤erent coordinate system, eddy transport of momentum can look
quite di¤erent, physically as well as mathematically.
2.3
2.3.1
QG momentum, heat, and PV transport
The quasigeostrophic equations on a -plane
We consider motion with characteristic horizontal length scale L, height scale h, 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 isentropic slopes j x j=j z j and j y j=j z j are
would not be small), and
3
Ro (h=L), (otherwise vertical motions
Under the “shallow atmosphere" approximation which all these equations assume, the slope is small, so
@z=@x = tan ' sin :
4
(iv) the static stability z is, to leading order, a function of z only [this stems from (iii)],
so we write = 0 (z) + ~, where ~ is O(Ro) smaller than 0 . We then de…ne a
background geopotential, in hydrostatic balance with the background 0 :
z
=
g
cp T
0
=
0
H
:
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
y
;v=
x
;w=0;
(14)
is the geostrophic streamfunction, which we de…ne to be
(z)] =f0 .
=[
(15)
Hydrostatic balance then gives us
z
=
1
f0
f0 H
z
=
f0 H
[
0 (z)]
=
f0 H
~ .
(16)
At next order, we obtain our quasigeostrophic equations. The equations of motion are
Dg u
yv f0 va = G (x) ,
Dg v + yu + f0 ua = G (y) ,
(17)
where G represents frictional or other forces per unit mass, Dg is the time derivative following
the geostrophic ‡ow
Dg @t + u@x + v@y
and (ua ; va ; wa ) is the ageostrophic velocity (i:e:, the di¤erence between the actual velocity
and the geostrophic one). Similarly, the thermodynamic equation becomes
Dg + wa
0;z
1
= cp
Q=(
)
1
J ,
(18)
where J is the diabatic heating rate per unit volume (and Q = J = cp is the heating rate
expressed in units of K per unit time). From these, we can readily derive the equation for
quasigeostrophic potential vorticity, q :
Dg q = X ,
where
q = f0 + y + vx
uy +
~
f0
0;z
5
(19)
!
z
= f0 + y +
2
(20)
and
X = Gx(y)
where
f0
Gy(x) +
,
0;z
(21)
z
f02 @
N 2 @z
@2
@2
1 @
+
+
2
2
@x
@y
@z
2
J
1
,
(22)
with
N2 =
0;z
H
being the square of the buoyancy frequency. Eq. (19) tells us that, for conservative ‡ow
(G = 0, J = 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.
2.3.2
PV ‡uxes and the Eliassen-Palm theorem
Consider perturbations to a steady, zonally-uniform basic state
U = U (y; z) ;
=
(y; z) ;
=
(y; z)
where
y
=
U;
y
H
= f0 Uz .
The basic state PV is:
Q(y; z) = f0 + y +
yy +
2
= f0 + y +
f02
N2
1
z
z
.
(23)
Write
=
where
0
+
is a small perturbation. Now, v 0 =
q0 =
0
2
=
0
xx
0
(x; y; z; t)
0
x
and
0
yy
+
+
f02
N2
1
0
z
;
(24)
:
(25)
z
therefore
v0q0 =
0
x
0
xx
+
0
x
0
yy
+
1
0
x
Simple manipulation yields the following identities:
6
f02
N2
0
z
z
1.
0
x
0
xx
2.
0
x
0
yy
1h
= (
2
0 2
x)
=
0
x
0
y y
=
0
x
0
y y
=
0
x
0
y y
;
0
z
i
=0;
x
0
y
1h
2
0
xy
0 2
y
i
x
and
0 1
x
f02
N2
0
z
=
1
f02
N2
0
x
1
f02
N2
0
x
1
f02
N2
0
x
z
=
=
f02
N2
z
f02 h
(
2N 2
0
z
z
0
z
0
z
0
xz
0 2
z)
i
x
:
z
Therefore, from (25),
v0q0 = r F .
where
F=
F (y)
F (z)
=
0
x
f02
0
x
0
y
0
2
z =N
!
(26)
=
u0 v 0
f0 v 0 0 =
:
(27)
0z
F is known as the ELIASSEN-PALM ‡ux. Note that the northward component of F is
(minus) the northward eddy ‡ux of zonal momentum, while the vertical component is proportional to the northward eddy ‡ux of heat, v 0 T 0 :
Now, assume that the perturbations are small, such that the QGPV equation (19) can
be linearized to give
(@t + U @x ) q 0 + v 0 Qy = X 0 ;
multiply by q 0 and average:
If we de…ne
A=
1 02
q
2
t
+ v 0 q 0 Qy = v 0 X 0 :
1 02
q =Qy and
2
D = v 0 X 0 =Qy ;
7
(28)
then, using (26),
At + r F = D .
(29)
Eq. (29) 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
At + r c g A = D .
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 Qy 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 (29).]
2.3.3
The Eliassen-Palm theorem
For waves which are steady (At = 0), of small amplitude and conservative (D = 0) , the ‡ux
F is nondivergent. This is, through (26), 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 (28) without involving F]:
2.3.4
Potential vorticity transport and the nonacceleration theorem
We now consider the problem of how eddies (in this case, quasigeostrophic eddies) impact
the zonal mean circulation. We return to eddies which may not be small and to the QGPV
budget, which becomes, on zonal averaging
qt + (v 0 q 0 )y = X .
(30)
Note that, unlike (5), there is no mean advection term in (30). 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, because of QG scalings.
8
The dynamical in‡uence of the eddies on mean potential vorticity, therefore, is described
entirely4 by the northward ‡ux v 0 q 0 . Given the equivalence between pv 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
= qt = X :
If qt 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. 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 =
= (f0 T =g ) z , it
y and
then follows that, under conditions (I)-(IV), the tendencies of the mean geostrophic ‡ow,
mean temperature and mean QGPV are independent of the waves. This is known as the
nonacceleration theorem and conditions (I)-(IV) are sometimes known as “nonacceleration conditions”.
2.3.5
Zonal mean momentum and heat budgets: Conventional Eulerian-mean
approach
In a similar fashion, we can take the zonal mean of the quasigeostrophic momentum and
heat equations. The zonal mean of the …rst of (17) gives us
ut
f0 va = G
(x)
4
(u0 v 0 )y ,
(31)
One should add the caveat that, in principle, the eddies could also in‡uence X (e.g., eddies could modify
mean precipitation).
9
where we have used v = x = 0 (the mean ageostrophic wind is not zero, however). Similarly,
the heat equation (18) gives
@
+ wa
@t
0z
=(
)
1
J
(v 0 0 )y .
(32)
The mean ‡ow equations are closed by the continuity equation
1
va;y + ( wa )z = 0
(33)
and the thermal wind equation
~y .
(34)
H
(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 (31) - (34). 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 heat v 0 0 (strictly, the eddy ‡ux
of heat is cp v 0 T 0 , but we shall refer to v 0 0 in this way for simplicity). Both these terms
force the mean ‡ow equations and it is important to note that the whole system is coupled,
i:e:, the heat ‡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, an upward
propagating wave with v 0 0 6= 0 but u0 v 0 = 0. The mean state could not respond with a
changing mean temperature only; thermal wind balance demands a corresponding change in
u. From (31) this could only be achieved through an ageostrophic meridional circulation,
which would impact on both the momentum and heat budgets. Thus, the waves will not
only drive ut and 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 (31) - (34) give us is a set of 4
equations in the 4 unknowns ut , 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” ut and 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 (31) and (32) exactly balance the eddy ‡ux terms. All in all,
this approach to the mean momentum and heat budgets is not giving us much insight into
what is going on.
f0 uz =
10
2.4
Transformed Eulerian-mean theory
The most important lesson to learn from the isentropic-coordinate analysis we did earlier is
that the de…nition of “mean,”and consequently of concepts like eddy ‡uxes, is non-unique.
We can take this message to heart by allowing ourselves some ‡exibility in de…ning what we
mean by the mean meridional circulation.
We begin by noting that, from (33), we may de…ne an ageostrophic mean streamfunction
a such that
1 @ ( a)
@ a
(va ; wa ) =
,
.
(35)
@z
@y
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
1@(
)
,
@z
(v ; w ) =
@
@y
(36)
;
where the new streamfunction is
=
+
a
c:
If we substitute this into (32), we obtain
@
+w
@t
Noting that
0
=
0 (z),
0z
=(
@ 0 0
(v )
@y
J
0z
@ c
.
@y
it follows that if we make the choice
c
so that
1
)
0
v0
=
,
0z
— the streamfunction of the so-called residual circulation— is
=
v0
a
0
,
(37)
0z
we obtain the thermodynamic equation
@
+w
@t
0z
=(
)
1
J .
(38)
We have thus succeeded in deriving a mean heat equation in which there are no explicit
eddy terms; heat 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 heat budget into its simplest possible form.
11
We now need to complete our transformed system of equations. The continuity equation
is
@v
1@( w )
=0
+
@z
@y
(39)
[we arranged this by (36)]. The thermal wind equation stays as before, i:e: :
f0 uz =
H
~y :
(40)
The momentum equation is less trivial. We need to replace va using (36); the result is
@u
@t
1
f0 v = G x + r F ,
(41)
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 (38) - (41), a
set of equations for v ; w ; @u=@t and @ =@t in which the only term representing eddy forcing
is a term 1 r F, 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 @ =@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 1 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. Moreover, through the relationship (26) we immediately
recover the central role of PV ‡uxes in the eddy forcing of the mean state.
2.4.1
Relationship to isentropic mean perspective
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 is rather more than that. It might
seem odd to have what looks like an eddy heat ‡ux appearing in the vertical momentum ‡ux
in this formulation. But it can be shown (see relevant Problem Set) that, for small-amplitude
eddies under adiabatic conditions:
12
1. this heat ‡ux term corresponds to the form stress term that appeared in the isentropic
coordinate formulation, and
2. that the “residual circulation” corresponds to the density-weighted mean circulation
is isentropic coordinates. [It is also, under these assumptions, directly related to the
Lagrangian mean motion.]
Thus, aside from the mathematical advantages of the TEM set of equations over the
‘’conventional” set, the formaultion is also describing the physics rather well. [You might
well ask: why not use the isentropic coordinate version? The answer is that, for wave
problems, isentropic coordinates are a bit unwieldy.]
2.4.2
Boundary conditions on the TEM equations
Finally, a short but important note about boundary conditions on the TEM ‡ow. At
a ‡at lower boundary, the boundary condition w = 0 translates straightforwardly, under
conventional averaging, to w = 0. For the TEM equations, however, (36) and (37) tells us
that
!
@ v0 0
w =
@y
0z
which is nonzero whenever there is a heat ‡ux along the boundary. So, in this respect, the
TEM framework is the more complicated (though it is actually telling us something about
the dynamics near the boundary — see Problem Set).
As we shall see, on a lower boundary with topography, the opposite may be true. In
general w is nonzero on such a boundary.
13