Potential Momentum and the equilibration of short Charney waves
by
Pablo Zurita-Gotor∗ GFDL/UCAR Visiting Scientist Program. Princeton, NJ 08542 and Richard S. Lindzen Program in Atmospheres, Oceans and Climate, MIT, Cambridge, MA 02139
Submitted to Meteorologische Zeitschrift Submitted: September 2005 Revised: March 2006
∗
Corresponding author address: Dr. Pablo Zurita-Gotor GFDL, Rm 237
US Route 1. Princeton, NJ 08542 Email:
[email protected]
Abstract
This paper proposes a new formalism for studying the extratropical circulation, based on the concept of potential momentum. Mathematically, the potential momentum is defined as the zonal momentum profile that produces the same potential vorticity distribution as the baroclinic term. Physically, easterly (westerly) potential momentum is associated with isentropic layers that open up (close down) with latitude. The former is the case in the extratropical troposphere, and gives a negative contribution to the interior potential vorticity gradient. Likewise, the surface temperature variance is interpreted in this framework as a reservoir of easterly potential momentum at the surface. With this redefinition of the mean flow, the wave-mean flow interaction problem can be recast in a form that is reminiscent of the barotropic framework. In particular, the eddy PV flux is the only wave-mean flow interaction term, and exchanges eddy pseudomomentum and mean flow momentum (which includes both the standard momentum and potential momentum) locally. On the other hand, the partition of the mean flow momentum between its physical and potential components is determined remotely, subject to the thermal wind constraint. This constraint is enforced by a residual circulation that exchanges both terms without affecting the total momentum or the full PV gradient. The potential momentum framework relates thermal homogenization at the surface to the net interior eddy absorption, regardless of whether this results from a meridional or a vertical EP convergence. This helps explain previous results by the same authors that a limited interior PV gradient does not constrain thermal homogenization at the surface, since in the 3D problem the waves can also refract meridionally to be absorbed at other latitudes.
1
1
1.
Introduction
2
In a recent two-paper series (Zurita-Gotor and Lindzen, 2004a, b), we have discussed the equi-
3
libration of short Charney waves, with the goal of understanding what prevents the eddies from
4
eliminating the extratropical surface temperature gradient. Since it is this temperature gradient
5
that provides the required sign change for instability (Charney and Stern, 1962), this is a central
6
question for theories of baroclinic adjustment, which assume a neutral mean state (see Zurita-
7
Gotor and Lindzen (2006b) for a review). A possible explanation was given by Lindzen (1993),
8
who proposed the Eady problem at the shortwave cutoff as the paradigm for the neutralized state.
9
However, it was noted by Zurita and Lindzen (2001) that the degree of PV homogenization in the
10
extratropical troposphere is comparable in the interior and at the surface, and that the interior PV
11
gradient only vanishes across a shallow region surrounding the steering level. These authors also
12
showed that short Charney waves (i.e., waves that see a small interior PV gradient) could also be
13
neutral when the PV gradient vanishes at the steering level alone.
14
Motivated by that analysis, Zurita-Gotor and Lindzen (2004a) investigated the equilibration of
15
short Charney waves in the 2D case, using the barotropic analog to the Charney problem proposed
16
by Lindzen et al. (1983). Their results supported the hypothesis that the baroclinic problem could
17
equilibrate through partial PV homogenization at the steering level. However, Zurita-Gotor and
18
Lindzen (2004b, ZL04 hereafter) found that in the 3D problem, this was only the case in the presence
19
of sufficiently strong surface friction. Otherwise, the interior PV gradient is greatly reinforced by
20
the meridional curvature of the enhanced barotropic jet and is no longer limited as for a short
21
Charney mode. ZL04 interpreted these results in terms of the full two-dimensional (in y and z)
22
redistribution of the zonal-mean momentum. The authors argued, based on the association of the
23
Eliassen-Palm (EP) flux F with a flux of easterly pseudomomentum (Edmon et al., 1980), that
24
the baroclinic equilibration may be interpreted in terms of the export of easterly momentum away 2
25
from the surface baroclinic zone. While in the purely baroclinic case this momentum can only be
26
transferred vertically (and might thus be limited by the depth of the modes), in the full 3D case
27
this constraint no longer applies because the easterly momentum can also be transferred laterally.
28
ZL04 implicitly assumed that the EP flux convergence ∇ · F = v 0 q 0 was associated with a local
29
modification of the mean flow. However, this association is not necessarily justified because, al-
30
though v 0 q 0 encapsulates the net wave-mean flow interaction, the mean flow responds in a non-local
31
manner to this forcing. To fix ideas, consider the 2D zonal momentum balance first: ∂U + αM U = v 0 ξ 0 ∂t
(1)
32
where αM is a frictional time scale. As discussed by Zurita-Gotor and Lindzen (2004a), the local
33
balance in equilibrium between the eddy vorticity flux and the mean flow forcing implies that the
34
equilibrated flow must maintain a negative (positive) PV gradient over those regions where the mean
35
flow acceleration is westerly (easterly), so that the eddy vorticity flux (assumed downgradient) can
36
support the mean flow imbalance against friction. In contrast, in the 3D case
37
∂U + αM U = f 0 v ∗ + v 0 q 0 ∂t 38
there is not such a local balance because the mean flow response involves a vertical redistribution
39
of the eddy forcing by the residual circulation. Hence, it is not possible in principle to associate
40
this eddy forcing with a local modification of the mean flow.
41
However, it will be shown in this paper that a generalization of the barotropic relation Eq. 1 can
42
still be obtained in the 3D case using the concept of potential momentum recently introduced by
43
Zurita-Gotor and Lindzen (2006a, ZL06 hereafter). This generalization provides the theoretical
44
justification for the picture proposed above. In particular, the surface temperature variance is
45
reinterpreted as an easterly momentum source, depleted by the surface eddy heat flux, while the 3
46
interior PV flux appears as a local easterly forcing for the mean flow. The theoretical framework is
47
presented in section 2, which essentially follows a parallel derivation in ZL06. Section 3 illustrates
48
these concepts by presenting some new diagnostics for the problem of equilibrating short Charney
49
waves. Section 4 concludes with a brief summary.
50
2.
51
Formulation
52
We start with the quasigeostrophic Boussinesq zonal-mean equations on the β plane:
The concept of potential momentum
∂U ∂ 0 0 + u v − f0 v a = −αM U ∂t ∂y ∂θ ∂ 0 0 v θ + w a Θz = −αT (θ − θ R ) + ∂t ∂y ∂v a ∂wa + =0 ∂y ∂z g ∂θ ∂U =− ∂z f0 Θ0 ∂y
(2) (3) (4) (5)
53
where the subscript a stands for ageostrophic variables. α M , αT are the linear damping coefficients
54
for Rayleigh friction and Newtonian cooling, and θ R is the radiative equilibrium profile to which
55
temperature is relaxed. We will assume α T to be constant, while αM is allowed to be height-
56
dependent (for instance, it may vanish above the boundary layer). Finally, Θ z is the reference
57
stratification of qg theory, which may also be height-dependent, and θ represents differences from
58
that reference state.
59
We will rewrite the thermodynamic equation in the form of a momentum equation so as to put
60
it into a form similar to that of Eq. 2, the momentum equation. We multiply Eq. 3 by −f 0 /Θz ,
61
integrate meridionally between y0 and y and differentiate with respect to z. Assuming that a 4
62
latitude y0 = 0 exists, such that v a = v 0 = 0 at y0 , the thermodynamic equation can then be
63
written: ∂ ∂M − ∂t ∂z
64
f0 0 0 v θ + f0 v a = −αT (M − M R ) Θz
(6)
where M =−
Z
y 0
∂ ∂z
f0 θ dy 0 Θz
(7)
65
is defined to be the potential momentum.
66
Note that M has momentum units, and that Eq.6 looks like a momentum equation. In particular,
67
the second term in that equation is interpreted in the Eliassen-Palm formalism as a vertical eddy
68
momentum flux (Edmon et al., 1980). This term can also be rewritten:
69
∂ − ∂z
f0 0 0 ∂ v θ = −v 0 Θz ∂z
f0 0 ∂m0 θ = v0 Θz ∂y
(8)
70
where m0 is the eddy component of the potential momentum M , and we took into account the
71
thermal wind relation for v 0 and the definition,
72
written in Transformed Eulerian Mean (TEM) form:
∂M ∂y
∂ = − ∂z
f0 Θz θ
. Eqs. 2, 6 can alternatively be
73
∂ ∂z
v0 θ0 Θz
∂U − f0 v ∗ = v 0 q 0 − αM U ∂t
(9)
∂M + f0 v ∗ = −αT M − M R ∂t
(10)
74
where v ∗ = v a −
75
Comparing Eqs. 9 and 10 (or Eqs 2 and 6), we can see that the residual/mean meridional circulation
76
simply converts potential momentum to physical momentum. We can eliminate this conversion term
77
by adding either set of equations together to obtain:
is the residual meridional velocity.
∂ ∂ 0 0 ∂ f0 0 0 U + M + αT M − M R + αM U = vθ − u v = v0 q0 ∂t ∂z Θz ∂y
5
(11)
78
As can be seen, in this formulation the forcing by the mean meridional circulation disappears, so
79
that the only dynamical forcing is the eddy PV flux v 0 q 0 . Hence, this equation can be regarded as
80
a generalization of the barotropic momentum equation Eq. 1. The main difference is that, in this
81
case, the eddy PV flux forces what we call the total momentum, which also includes the potential
82
momentum.
83
Eq. 11 can also be combined with the conservation of pseudomomentum: ∂ (M + U ) = v 0 q 0 + Dmean ∂t ∂ A = −v 0 q 0 + Deddy ∂t
(12) (13)
84
where A ≈ q 02 /2q y (see e.g., Andrews et al. (1987)), D mean = −αT (M − M R ) − αM U is the
85
mean flow forcing and the net eddy dissipation D eddy includes frictional, thermal and small scale
86
contributions (see ZL06 for details). These equations emphasize that there can only be a wave-
87
mean flow interaction when the eddy PV flux is non-zero (Charney and Drazin, 1961). For the
88
(time-mean) equilibrated state: −Dmean = v 0 q 0 = Deddy
(14)
89
which reflects a local balance everywhere between the forcing of the mean flow, the eddy-mean
90
flow interaction and the eddy dissipation. Hence, the mean flow is only adjusted (in the sense that
91
Dmean 6= 0) over those regions with non-zero PV flux. Based on the wave-geometry interpretation of
92
baroclinic instability (Lindzen et al. (1980), Lindzen (1988)), we expect the bulk of this adjustment
93
to occur at the surface and steering level (see also section 3).
6
94
Non-locality and relation to wave propagation
95
A conceptual advantage of this framework is that the total momentum M + U is forced locally by
96
the eddy PV flux (c.f., Eq. 12). In contrast, when only U is considered the mean flow response
97
to the forcing is non-local because the residual circulation spreads the forcing vertically. However,
98
note that M , as defined in 7, is a non-local function of the basic state, which depends on the
99
full isentropic structure equatorward of the given location. Hence, the non-local character of the
100
original equations has in a sense only been hidden in the definition of the basic state. Additionally,
101
although M + U is forced locally, its partition is determined remotely. At any time, this partition
102
must be such that thermal wind balance is satisfied: ∂ ∂2M = 2 ∂y ∂z
f02 ∂U N 2 ∂z
!
(15)
103
This constraint is enforced by the residual circulation, which exchanges both forms of momentum
104
according to Eqs. 9-10.
105
Non-locality should not be surprising since baroclinic instability is fundamentally associated with
106
the interaction of propagating waves with the mean flow (Lindzen, 1988). The fact that the eddy
107
PV flux may also be written as the divergence of a wave flux (Edmon et al., 1980) implies that
108
it cannot be determined locally: indeed, it has to integrate globally to zero (Bretherton, 1966).
109
Though Eq. 14 suggests a three-way local balance between the mean flow forcing, the wave-mean
110
flow interaction and the eddy dissipation, the locality of this balance only holds for the time-mean.
111
The adjustment of the basic state that defines D mean is still non-local.
7
112
Isentropic interpretation
113
The concept of potential momentum is somewhat similar to that of available potential energy.
114
While the concept of available potential energy exploits the sloping of the isentropes, the concept
115
of potential momentum is based on the non-uniformity of the isentropic thickness. As implied by
116
its name, M represents the zonal momentum that would be realized if the basic state stratification
117
were brought to its reference value Θ z while conserving the same potential vorticity distribution.
118
This is evident from its definition: ∂ ∂M =− ∂y ∂z
119
f0 θ Θz
which also allows us to rewrite the quasigeostrophic PV gradient: qy = β − ∂yy U + ∂yz (
f0 θ) = β − ∂yy (U + M ) Θz
(16)
120
For a stratified rotating fluid, an important component of the basic state potential vorticity is built
121
into the stretching term
122
would produce a meridional wind shear
123
which would result from bringing the isentropic thickness down to its reference value (which may
124
be height-dependent) at all latitudes poleward of y 0 .
125
This is illustrated in Fig. 1, which shows two idealized distributions of the isentropic thickness for
126
a generic isentropic layer θi , together with the corresponding distribution of M . Also shown is the
127
reference thickness (dashed). When the isentropic thickness is larger than in the reference state
128
(i.e.,
129
its reference value, the flow would generate an enhanced meridional westerly shear so as to decrease
130
its absolute vorticity and maintain constant potential vorticity. The reverse is also true, so that M
131
decreases with latitude when the isentropic thickness is smaller than in the reference state.
∂θ ∂z
∂ ∂z
f0 Θz θ
. If this term were entirely converted into relative vorticity, it ∂M ∂y .
Hence, M gives the net change in zonal momentum
< 0), M must increase with latitude. In other words, if the thickness were brought down to
8
132
For the sketch shown in the left panel, the thickness increases with latitude monotonically. If the
133
reference stratification is properly defined, this implies that the thickness must be smaller than
134
the reference thickness for the first half of the profile, and larger for the second. Consequently, M
135
initially decreases with latitude, and increases near the end. Taking into account that M (y 0 ) = 0,
136
this produces an M profile as shown. Moreover, when the mean stratification agrees with the
137
reference stratification
138
The reverse is true for the case shown in the right panel, for which the thickness decreases mono-
139
tonically with latitude. Hence, we can see that an easterly (westerly) potential momentum M is
140
associated with isentropic layers that open up (close down) with latitude, as represented schemat-
141
ically in Fig. 1. When the dependence of
142
Because of the boundary conditions M (y 0 ) = M (yL ) = 0, negative (positive) midlatitude values of
143
M are typically associated with positive (negative) values of M yy . From equation 16, this implies
144
that the M contribution to the PV gradient is negative (positive) when M is easterly (westerly). In
145
the troposphere, the slope of the isentropes increases with height (Stone and Nemet, 1996), which
146
implies that the isentropic layers open up with latitude and M is easterly. This gives a negative
147
contribution to the interior PV gradient, as discussed by Zurita and Lindzen (2001).
148
We can also interpret the generation of M in terms of the isentropic view put forward above. Eq.
149
10 shows that potential momentum can only be generated diabatically or converted adiabatically
150
from/into U . In the interior, this adiabatic conversion occurs through a poleward residual circula-
151
tion, that thins (thickens) the isentropic layers at low (high) latitudes. According to Fig. 1, this
152
generates easterly potential momentum, at the expense of an equal westerly U acceleration via the
153
Coriolis force. The opposite conversion occurs at the surface, where the return flow is equatorward.
154
The diabatic generation can be interpreted in a similar manner. Differential heating adds mass
155
to the warmer isentropic layers at low latitudes, and to the colder layers at high latitudes. For a
∂θ ∂z
has zero mean and M (yL ) = 0 as well at the right endpoint yL .
∂θ ∂z
on latitude is linear, M is quadratic in y.
9
156
stably stratified fluid, this implies that the lower troposphere isentropic layers are thinned at low
157
latitudes and thickened at high latitudes, whereas the opposite is true for the upper tropospheric
158
layers. Based on Fig. 1, this can be interpreted as a generation of easterly (westerly) potential
159
momentum at lower (upper) levels.
160
Surface potential momentum
161
Following Bretherton (1966), we can substitute the surface boundary condition for an isothermal
162
surface underneath a temperature jump. This temperature jump produces a delta-function poten-
163
tial momentum jet, as can be easily derived from Eq. 7: M (y, z) = −
f0 Θz
Z
y
θ δ(z) dy 0
at z = 0
(17)
0
164
where the temperature of the isothermal surface has again been subtracted. Since this temperature
165
can be arbitrarily chosen, we choose it to be the mean surface temperature between y 0 and yL ,
166
which makes the surface M vanish at both endpoints. Eq. 17 implies that a negative temperature
167
gradient as observed is equivalent to an easterly potential momentum reservoir. This also gives a
168
negative delta-function contribution to the PV gradient (c.f., Eq. 16).
169
An important consequence of the non-local definition of M is that its value may be sensitive to
170
the choice of endpoints y0 , yL . In the interior, this dependence disappears as long as y 0 is chosen
171
outside the adjusted region (assuming that the unperturbed stratification equals the reference
172
stratification, which is reasonable in qg theory). However, this is not the case at the surface because
173
the surface temperature at large positive and negative y is generally different from the mean surface
174
temperature, even if the far-range temperature gradient vanishes. As a result, the surface value of M
175
is ambiguous. This is discussed extensively by ZL06, who show that the ambiguity disappears when
176
the potential momentum correction M − M R is considered and y0 is chosen outside the adjusted 10
177
region. In other words, the choice of y 0 carries an implicit assumption on the meridional extent of
178
the domain over which the thermal field can be rearranged. ZL06 show that, in many instances,
179
differences in the surface potential momentum correction are due to the meridional expansion of
180
the adjusted domain. For this reason, they argue that potential momentum gives a more complete
181
measure of the eddy source than local instability measures, such as baroclinicity.
182
Another important result of ZL06 is that the vertically integrated potential momentum is conserved: Z
∞
M dz = 0
Z
∞ 0
M R dz
(18)
183
where MR is the radiative equilibrium potential momentum, and the integrals include generalized
184
delta-function contributions at both boundaries. Moreover, each of the terms in Eq. 10 integrate
185
independently to zero. The easterly generation of interior M by the residual circulation is accompa-
186
nied by an equal generation of westerly M at the surface, associated with the return flow. Likewise,
187
the diabatic generation of easterly potential momentum at lower levels is compensated by an equal
188
generation of westerly M aloft.
189
General circulation
190
The general circulation can be described in terms of potential momentum as follows.
191
1. Diabatic processes generate easterly potential momentum at lower levels, and compensating
192
westerly potential momentum aloft. In particular, they force the surface temperature gradi-
193
ent, which can be interpreted as a reservoir of easterly potential momentum at the ground.
194
2. The surface easterly potential momentum is equivalent to a negative delta-function PV gradi-
195
ent which is needed for the existence of unstable baroclinic modes (Charney and Stern (1962), 11
196
Lindzen (1988)). The equilibration of the baroclinic waves produces downgradient PV fluxes,
197
thus positive (negative) PV fluxes at the surface (in the interior).
198
3. These eddy PV fluxes are balanced by a direct circulation v ∗ . The poleward interior branch
199
transforms (westerly) potential into physical momentum. The delta-function return flow de-
200
pletes the easterly reservoir of potential momentum/reduces the surface temperature variance.
201
4. The system reaches an equilibrium when the diabatic generation of westerly potential mo-
202
mentum in the interior is balanced by the easterly drag, and both are equal to the rate of
203
conversion from M into U by the poleward residual circulation.
204
In the time-mean, Eq. 14 represents a local balance everywhere between the restoration of the
205
mean flow, the eddy dissipation, and the wave-mean flow interaction. Although this is a powerful
206
constraint, in many instances the global integral of these quantities might be of more relevance.
207
For instance, a global value of Dmean provides an integral measure of the mean flow adjustment,
208
which would seem more relevant than its value at any given point. However, the main difficulty
209
for constructing global balances is the fact that the quantities in Eq. 14 are not sign-definite. In
210
particular, it is well known (Bretherton, 1966) that v 0 q 0 integrates globally to zero; Eq. 14 implies
211
that so must Dmean and Deddy in equilibrium. As a result, the global integral of Eq. 14 trivially
212
vanishes.
213
For this reason, it is more meaningful to consider the integral of these terms over the regions where
214
they are positive (which also equals minus the integral over the region with negative values, or
215
half the global integral of the absolute values). This is not merely a mathematical convenience,
216
but also physically meaningful. The association of the eddy PV flux with the divergence of a wave
217
flux v 0 q 0 = ∇ · F (Edmon et al., 1980) allows one to associate the regions with positive/negative
218
PV fluxes with a wave source/sink. Thus, the positive integral may be regarded as the net wave 12
219
source, which must also equal the net wave sink. Assuming that the PV fluxes are downgradient in
220
the time-mean, the separation between the regions with positive/negative PV fluxes is in practice
221
equivalent to the separation between the regions with negative/positive PV gradients, or the surface
222
and the interior troposphere (Held, 1999).
223
Thus, we define the intensity of the global circulation (following ZL06): C=−
Z Z
sur
Dmean =
Z Z
sur
αT (M − M R ) =
Z Z
int
Dmean = −
Z Z
v0 q0 = − int
Z Z
int
Deddy = ... (19)
224
where ’sur’ and ’int’ stand for the surface 1 and interior regions respectively. These integrals are
225
also equal to the net mass flux
226
As discussed above, we can associate the surface integral of D mean with the net wave source. On
227
the other hand, the potential momentum depletion:
RR
f0 v ∗ over each region (c.f., Eq. 10).
Z=−
Z Z
sur
(M − M R ) = C/αT
228
can be regarded as a measure of the thermal homogenization at the surface, i.e., how much the
229
surface temperature variance is reduced from radiative equilibrium. Eq. 19 shows that this term
230
is also related to the net eddy dissipation in the interior, which suggests that enhanced interior
231
dissipation might lead in some cases to enhanced thermal homogenization at the surface. Section
232
3 confirms this for a simple case. 1
For the argument discussed here, ’surface’ refers to the massless layer of qg theory (the delta-function). In an
actual multilayer/numerical model the surface would have a finite thickness, so that frictional contributions would also be lumped into the surface Dmean . For example, see ZL06 for the two-layer case.
13
233
3.
Some simple examples
234
Short Charney waves: a 3D view
235
In this section we illustrate the concepts introduced above for a very simple problem: the equili-
236
bration of short Charney waves. These are modes for which the net PV gradient in the interior is
237
smaller than the integrated delta function at the surface (Held (1978), Zurita and Lindzen (2001)).
238
Because these modes do not grow as fast as the most unstable mode, the latter would always
239
dominate in a seminfinite fluid. However, short Charney modes may still be relevant when longer
240
modes are prevented by the geometry, for instance if the meridional scale imposed by the jet con-
241
strains the depth of the modes (Lindzen, 1993). Zurita and Lindzen (2001) express the scale of
242
the modes in terms of the dimensionless parameter H/h, where H is the half Rossby depth and
243
h=
244
the ratio between the interior and boundary PV gradient. Short Charney modes are characterized
245
by H/h < 3.9, which is the value corresponding to the most unstable mode.
246
When the interior PV gradient is small, the condition that the PV fluxes integrate vertically to
247
zero requires a large interior diffusivity. Zurita and Lindzen (2001) show that this implies that
248
the modal PV flux must be strongly-peaked at the steering level because the linear diffusivity is
249
bounded away from it (Bretherton, 1966). This is also consistent with the generalized instability
250
mechanism of Lindzen (1988) in which eddy growth is due to stimulated wave emission by means of
251
the Kelvin-Orr mechanism in the neighborhood of the steering level (or by wave over-reflection as it
252
is referred to in Lindzen et al. (1980)). For weakly growing modes, this implies that the wave-mean
253
flow interaction is limited to a narrow region surrounding the steering level. However, the region
254
with non-zero PV flux broadens as ci increases (Lindzen et al., 1980), which may be interpreted in
255
terms of the time-dependent forcing at the steering level (Charney and Pedlosky, 1963).
2 1 f0 dU β N 2 dz
z=0+
is the Held scale (Held, 1978). This parameter may be regarded as a measure of
14
256
Consistent with these ideas, Zurita-Gotor and Lindzen (2004a) show that, when only short Char-
257
ney modes are allowed, the inviscid 2D problem equilibrates by eliminating the PV gradient at
258
the steering level alone, while still keeping a negative PV gradient at equilibration. The authors
259
speculate that this might explain why the surface temperature gradient is not eliminated, and
260
the tropospheric PV gradient vanishes at the steering level alone, provided that some external
261
mechanism like the jet width constrains the scale of the modes. However, ZL04 find that these
262
constraints do not hold in the 3D case. The reason is that in the 3D problem momentum is not
263
only redistributed vertically, but also meridionally. In other words, the net interior PV flux is not
264
constrained by a small interior PV gradient aloft as in the 2D case because the waves can also
265
propagate meridionally to be absorbed at other latitudes.
266
In practice, ZL04 find that meridional propagation is most important during an initial adjustment
267
stage. As the waves propagate meridionally and the convergent momentum fluxes accelerate the
268
jet, the flow develops a large horizontal curvature. This curvature contributes to the positive PV
269
gradient above the surface (or more generally above the region with negative potential vorticity
270
gradient), so that the mode no longer behaves as a short Charney wave. As a result, the PV flux
271
broadens around the steering level and the surface vertical shear is eliminated. ZL04 find that this
272
process is largely controlled by friction. With strong surface friction, the flow cannot develop the
273
required meridional curvature and the wave remains ’short’. Thus, PV is only homogenized locally
274
at the steering level, as in the 2D case. ZL04 explain these results noting that, according to the
275
TEM framework, the equilibration can be interpreted in terms of the export of easterly momentum
276
away from the surface. In a purely baroclinic adjustment all this easterly momentum is deposited
277
aloft along the column, but in the 3D problem the lateral export is also important.
278
This is most transparent in the potential momentum framework since Eq. 19 shows that thermal
279
homogenization at the surface, as measured by the potential momentum, is related to the net
15
280
interior PV flux, not just the along-column baroclinic component. We next apply these diagnostics
281
to the same sets of runs discussed by ZL04. The model used is a qg model forced by linear
282
relaxation to a ‘radiative equilibrium’ Charney-like basic state that has constant vertical shear
283
with height. The radiative equilibrium zonal wind is meridionally modulated by a Gaussian jet
284
with halfwidth σ = 2000 km and has a maximum vertical shear Λ(0) = 33 ms −1 /10 km at the
285
jet center. The model has rigid lids at the top and meridionally, but at enough distance to be
286
considered in practice unbounded. A small channel length (L = 4000 km) is chosen to prevent the
287
dominance of the most unstable mode and to allow only short Charney waves. The dimensionless
288
scale of the first harmonic, measured by H/h, is changed through β as described in ZL04. All
289
simulations are run for 800 days starting from radiative equilibrium, with averages calculated over
290
the last 400 days. Additional details about the model are given in ZL04.
291
Fig. 2 shows some diagnostics for a run with H/h = 1.6 and frictional timescale 1 day. Since the
292
vertical shear is constant with height (i.e., θ y 6= f (z), or ∂y θz = 0), the radiative equilibrium profile
293
has zero potential momentum in the interior. In contrast, panel A shows that the equilibrated
294
flow has developed, at equilibration, an easterly jet of potential momentum. Consistent with the
295
isentropic view put forward in section 2, panel B shows that the easterly potential momentum is
296
associated with isentropes that open up with latitude. However, note that the (scaled) isentropic
297
slope is large and that the changes in thickness at constant height are larger than following the
298
actual isentropes. It is noteworthy that the potential momentum jet of panel A is also comparable
299
to the actual jet of zonal momentum, shown in panel C. This is as required by the thermal wind
300
constraint (Eq. 15). Note that this also implies (panel D) that the PV gradients associated to the
301
horizontal curvature of the jet and stretching term (i.e., the potential momentum curvature, Eq.
302
16) are comparable.
303
Fig. 3A shows the (delta-integrated) surface potential momentum at equilibration (thin, solid) and
16
304
in radiative equilibrium (thin, dashed). Also shown is the vertically-integrated potential momen-
305
tum in the interior from Fig. 2A (thick, solid), which agrees exactly with the difference between
306
the two previous curves (thick, dashed). This is what should be expected, as the interior potential
307
momentum is initially zero and Eq. 18 implies that the vertically-integrated potential momen-
308
tum is conserved. The baroclinic equilibration can thus be interpreted in terms of the vertical
309
redistribution of the easterly M originally locked at the surface.
310
As discussed in section 2, the actual values of potential momentum at the surface depend on the
311
choice of endpoints y0 , yL , and are therefore ambiguous. This is illustrated in panel B of Fig. 3,
312
which shows the same curves of panel A for a different choice of the refefence latitude y 0 . As can be
313
seen, this gives different values of the surface M , both at equilibration and in radiative equilibrium.
314
Nevertheless, the difference between both curves should be the same, and equal to the integrated
315
M in the interior, as long as y0 is chosen within the unperturbed region (the small differences
316
between both panels are due to the violation of this condition for panel B). The main difference
317
between panels A and B is merely one of interpretation. When choosing the reference latitude,
318
one is implicitly assuming the size of the domain over which temperature can be rearranged, which
319
defines the maximum intensity of the circulation. Though the actual circulation C = α T (M − M R )
320
is the same with both choices of y0 , the theoretical maximum αT M R corresponding to full potential
321
momentum depletion is larger with the broader domain in panel A. The waves ’appear less efficient’
322
in reducing the surface potential momentum in that case because they homogenize temperature
323
over a narrower domain than anticipated. ZL06 show some examples in which differences in the
324
degree of thermal homogenization are due to differences in the meridional extent of the adjusted
325
domain, rather than to differences in the maximum baroclinicity.
326
Note that the interior potential momentum jet in Fig. 2 has a well-defined peak at the steering
327
level. This is not surprising since Eq. 14 implies that the potential momentum restoration must
17
328
equal the eddy PV flux locally. Because friction is sufficiently large, the mode still behaves like
329
a short 2D Charney wave and the wave-mean flow interaction/mean flow adjustment has a lot of
330
structure at the steering level. In contrast, Fig. 4 shows the potential momentum diagnostics for a
331
case with reduced surface friction (5 days). As can be seen, in this case the potential momentum
332
jet (and also the PV flux, not shown) have a much broader structure. This occurs as the flow
333
develops a strong barotropic jet (panel B) and a large horizontal curvature: Panel C shows that
334
this curvature produces a positive PV gradient (dashed line) that is nearly 3 times as large as
335
before. As a result, the mode no longer behaves as a short Charney mode: the PV flux is broader
336
and the interior jet of potential momentum jet is also stronger. Since the vertically integrated M
337
is conserved, this also implies stronger potential momentum depletion/thermal homogenization at
338
the surface (panel D).
339
Interior eddy dissipation and surface thermal homogenization
340
We showed above that for short Charney waves the degree of thermal homogenization at the surface
341
is controlled by the interior eddy absorption. When friction is weak, thermal homogenization
342
increases because the reinforced meridional curvature supports enhanced interior absorption. To
343
further stress this point, we next describe the sentivity of the potential momentum depletion M −
344
MR on the interior absorption. With this purpose, we have designed a series of idealized experiments
345
in which artificial wave dissipation is added in the interior. We simply add a term of the form ∂q 0 q0 = − + ... ∂t τ
346
to the non-zonal part of the flow. Thus, the waves are damped but the forcing of the basic state is
347
unchanged. All the remaining parameters are chosen as in Fig. 2.
348
From a linear point of view, this damping term should reduce the eddy amplitudes and fluxes, due 18
349
to its stabilizing effect. Indeed, in the limit of large damping, the radiative equilibrium solution
350
would be stable. Fig. 5A shows the linear growth rate as a function of τ . The linear growth rate
351
displays sensitivity to τ only when this damping is faster than the diabatic timescale; however,
352
even then the effect is moderate unless τ goes below 5 days or so.
353
On the other hand, the arguments of previous sections suggest that adding a wave sink in the
354
interior could actually lead to a stronger circulation. Indeed, in the strict non-dissipative limit
355
the non-acceleration theorem tells us that there cannot be a mean circulation. Fig. 5B shows the
356
dependence of the circulation (c.f., Eq. 19) on the wave damping timescale. As can be seen, the
357
circulation increases for nearly all timescales and reaches a maximum for τ ≈ 5 days, beyond which
358
value the direct damping effect presumably becomes important. The wave sink effect is clearly
359
significant: the circulation increases by as much as a 55% for τ = 5 days, and even for τ = 1 day it
360
is still a 45% stronger than in the control run. We have also performed experiments in which the
361
linear damping is added at the lower boundary alone. In that case the circulation always decreases
362
with the damping (not shown).
363
The increase in the circulation with increased damping may be surprising at first. Does this imply
364
that the eddy amplitude also increases with damping? As shown in panel C of Fig. 5, the answer
365
depends on the norm. The net interior eddy PV flux must of course increase to balance the
366
enhanced Deddy . However, this increase is mainly due to enhanced correlation between velocity
367
and PV perturbations. Although there is a weak eddy enstrophy maximum for τ = 5 days, the
368
meridional velocity variance always decreases with the damping.
369
Finally, Fig. 6 shows the structure of the interior eddy PV flux for the cases with no damping,
370
with interior damping with timescale τ = 5, 15 days and with boundary damping τ = 15 days.
371
Interior damping makes the PV flux less structured, as if the wave were deeper, whereas boundary
372
damping has the opposite effect and produces a shallower PV flux at the steering level. These 19
373
results suggest that it is the ratio of boundary to interior absorption, rather than between their
374
respective PV gradients, that defines whether a wave is short or long. Both would be equivalent
375
with uniform timescales, as in the standard linear problem.
376
The strong enhancement of the circulation as the interior absorption is enhanced should not be
377
surprising, as the defining characteristic of a short Charney wave is precisely the relative weakness
378
of the interior PV fluxes. However, this is not a general result: for instance, in the two-layer runs
379
of ZL06 the circulation always decreases with interior damping 2 . Although Eq. 19 implies that
380
increasing the net interior eddy dissipation |D eddy | should lead to enhanced thermal homogeniza-
381
tion, it is not obvious that artificially reducing the dissipative timescale somewhere necessarily
382
results in enhanced net absorption. In particular, the possibility cannot be discarded that the
383
local enhancement in the wave sink occurs at the expense of weakening other sinks, rather than
384
through an enhancement of the wave source. A specific example of how damping contributes to
385
the destabilization of baroclinic modes is given in Snyder and Lindzen (1988).
386
4.
387
In this paper we have discussed the equilibration of a forced-dissipative baroclinic system using
388
the generalized momentum framework recently derived by Zurita-Gotor and Lindzen (2006a). In
389
this framework, the thermal structure is recast in a momentum form using the concept of potential
390
momentum. The potential momentum is simply the zonal momentum profile that produces the
391
same PV distribution as the stretching term. When the total momentum is considered, the eddy
392
PV flux is the only forcing of the mean flow, which may be regarded as the mean flow counterpart to
Conclusion
2
Note that in the two-layer model the positive PV gradient is always larger than the negative PV gradient, due
to the presence of an upper rigid lid.
20
393
pseudomomentum conservation. On the other hand, the partition between potential and physical
394
momentum is subject to the thermal wind constraint. This is enforced by the residual circulation,
395
which exchanges potential and physical momentum without changing the total momentum.
396
We have also interpreted the equilibration of short Charney waves, previously discussed by Zurita-
397
Gotor and Lindzen (2004 a, b), using these concepts. In the new framework, the thermal ho-
398
mogenization at the surface is measured in terms of the potential momentum depletion. Unlike
399
baroclinicity, this measure also accounts for changes in the meridional domain over which surface
400
temperature is rearranged. Since the potential momentum depletion increases with eddy absorp-
401
tion everywhere, not just aloft the baroclinic zone, the along-column PV gradient does not limit
402
thermal homogenization as in the 2D problem. Likewise, we found that interior eddy dissipation
403
leads to enhanced thermal homogenization at the surface for a short Charney mode, due to the
404
increase in the interior PV fluxes.
405
The framework considered here is very idealized. Besides the QG limitation, the assumption that a
406
quiescent latitude exists (or equivalently, that the domain is meridionally unbounded) is obviously
407
unrealistic for the Earth’s atmosphere. A more common setup in the QG context involves the use of
408
close meridional walls. As discussed by Zurita-Gotor (2006), this could have an important impact
409
on the equilibrated system by constraining the inverse cascade. Neither setup is very realistic, but
410
we believe that the cleaner unbounded setup discussed here might be a useful reference paradigm for
411
understanding the atmospheric case. One can easily overcome this limitation by including boundary
412
terms at the reference latitude, an approach that could be useful for studying the interaction
413
between the Hadley cell and the extratropical eddies within the qg framework. We intend to follow
414
this approach in a subsequent study.
415
To conclude, we note that the total momentum equation (Eq. 11) is simply the y-integral of the
416
zonal-mean quasigeostrophic potential vorticity equation. For symmetric 2D flow, momentum pro21
417
vides a full description and vorticity is redundant. Likewise, symmetric 3D flow can be encapsulated
418
in terms of momentum alone, provided that the baroclinic contribution to the potential vorticity
419
is also rewritten as a momentum term. While the present momentum formulation offers a certain
420
conceptual elegance, it remains to be seen whether it offers serious practical advantages. At the
421
very least, however, the explicit partition of PV between M and U allows one to better distinguish
422
and quantify the roles of vertical and meridional propagation.
423
ACKNOWLEDGMENTS: P.Z-G was supported by the Visiting Scientist Program at the NOAA
424
Geophysical Fluid Dynamics Laboratory, administered by the University Corporation for Atmo-
425
spheric research. R.S.L. acknowledges financial support by NSF Grant ATM-9421195 and DOE
426
Grant FG02-93ER61673.
22
427
References
428
Andrews, D. G., J. R. Holton, and C. B. Leovy: 1987, Middle Atmospheric Dynamics. Academic
429
430
431
432
433
434
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436
437
438
439
440
441
Press, 489 pp, first edition. Bretherton, F. P.: 1966, Critical layer instability in baroclinic flows. Q.J.Meteor.R.Soc., 92, 325– 334. Charney, J. G. and P. G. Drazin: 1961, Propagation of planetary scale disturbances from the lower into the upper atmosphere. J.Geophys.Res., 66, 83–110. Charney, J. G. and J. Pedlosky: 1963, On the trapping of unstable planetary waves on the atmosphere. J.Geophys.Res., 68, 6441–6442. Charney, J. G. and M. E. Stern: 1962, On the stability of internal baroclinic jets in a rotating atmosphere. J.Atmos.Sci., 19, 159–172. Edmon, H. J., B. J. Hoskins, and M. E. McIntyre: 1980, Eliassen-Palm cross-sections for the troposphere. J.Atmos.Sci, 37, 2600–2616. Held, I. M.: 1978, The vertical scale of an unstable baroclinic wave and its importance for eddy heat flux parameterizations. J.Atmos.Sci, 35, 572–576.
442
— 1999, The macroturbulence of the troposphere. Tellus, 51AB, 59–70.
443
Lindzen, R. S.: 1988, Instability of plane parallel shear flow (toward a mechanistic picture of how
444
it works). PAGEOPH , 126, 103–121.
445
— 1993, Baroclinic neutrality and the tropopause. J.Atmos.Sci, 50, 1148–1151.
446
Lindzen, R. S., B. F. Farrell, and K. K. Tung: 1980, The concept of wave overreflection and its
447
application to baroclinic instability. J.Atmos.Sci, 37, 44–63. 23
448
449
Lindzen, R. S., A. J. Rosenthal, and B. Farrell: 1983, Charney’s problem for baroclinic instability applied to barotropic instability. J.Atmos.Sci, 40, 1029–1034.
450
Snyder, C. and R. S. Lindzen: 1988, Upper level baroclinic instability. J.Atmos.Sci, 45, 2446–2459.
451
Stone, P. H. and B. Nemet: 1996, Baroclinic adjustment: a comparison between theory, observa-
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454
455
456
457
458
459
460
461
462
tions, and models. J.Atmos.Sci., 53, 1663–1674. Zurita, P. and R. S. Lindzen: 2001, The equilibration of short Charney waves: implications for PV homogenization in the extratropical troposphere. J.Atmos.Sci, 58, 3443–3462. Zurita-Gotor, P.: 2006, The relation between baroclinic adjustment and turbulent diffusion in the two-layer model. Submitted, J.Atmos.Sci. Zurita-Gotor, P. and R. S. Lindzen: 2004a, Baroclinic equilibration and the maintenance of the momentum balance. Part I: a barotropic analog. J.Atmos.Sci, 61, 1469–1482. — 2004b, Baroclinic equilibration and the maintenance of the momentum balance. Part II: 3D results. J.Atmos.Sci, 61, 1483–1499. — 2006a, A generalized momentum framework for looking at baroclinic circulations. Accepted, J.Atmos.Sci.
463
— 2006b, Theories of baroclinic adjustment and eddy equilibration, The global circulation of the
464
atmosphere: phenomena, theory, challenge. T. Schneider and A. H. Sobel eds. Princeton UP.
24
465
466
List of Figures 1
whose thickness: (left) increases with latitude, (right) decreases with latitude. . . . . . . . .
467
468
Sketch illustrating the potential momentum distribution associated to an isentropic layer
2
Potential Momentum Diagnostics for the run with H/h = 1.6 and frictional timescale 1
469
day. (A) Interior potential momentum in m/s; (B) Percentage change in the equilibrium
470
stratification with respect to the reference stratification Θz ; (C) Zonal wind in m/s; and
471
(D) Vertically-averaged interior PV gradient contributions by the M (solid) and U (dashed)
472
curvatures (normalized by β). The isentropes are shown dashed for reference in panels A-C.
473
3
26
27
(A) Surface potential momentum at equilibration (thin, solid) and in radiative equilibrium
474
(thin, dashed) for the run with H/h = 1.6 and frictional timescale 1 day. Also shown are
475
their difference (thick, dashed) and the integrated potential momentum in the interior (thick,
476
solid). (B) As A, but for a different choice of the reference latitude y0 . Units are in m2 /s. .
28 29
477
4
As panels A, C, D in Fig. 2 and panel A in Fig. 3, but with frictional timescale 5 days. . . .
478
5
As a function of the interior eddy damping timescale τ (A) Growth rate, (B) Net Circulation
479
(c.f., Eq. 19) and (C) Eddy amplitude in the indicated norms. The solutions in panels B
480
and C are normalized with the τ = ∞ solution. . . . . . . . . . . . . . . . . . . . . . . .
481
482
6
30
Interior eddy PV flux for the interior and boundary eddy damping timescales indicated. Units are in 10−5 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
31
θi
θi
Θz
yL
y0
yL
y0
M<0
Θz
M>0
Figure 1: Sketch illustrating the potential momentum distribution associated to an isentropic layer whose thickness: (left) increases with latitude, (right) decreases with latitude.
26
Potential Momentum
Stratification Adjustment
−8
z
z
−32
−16
24 32
−40
−32 −24
16
−24 8 −8
−16
A
B
y
y
Zonal Wind
PV Gradient
30
0.8 0.6
25
0.4 20
Myy, Uyy
0.2
z
15 10 5
0 −0.2 −0.4 −0.6 −0.8
5
−1
C
−1.2
y
M U
D y
Figure 2: Potential Momentum Diagnostics for the run with H/h = 1.6 and frictional timescale 1 day. (A) Interior potential momentum in m/s; (B) Percentage change in the equilibrium stratification with respect to the reference stratification Θz ; (C) Zonal wind in m/s; and (D) Vertically-averaged interior PV gradient contributions by the M (solid) and U (dashed) curvatures (normalized by β). The isentropes are shown dashed for reference in panels A-C.
27
5
2
5
x 10
2
x 10
1 1 0 0
−2
∫ M dz
∫ M dz
−1
−3 −4
−1
−2
−5 −3 −6
A −7
B −4
y
y
Figure 3: (A) Surface potential momentum at equilibration (thin, solid) and in radiative equilibrium (thin, dashed) for the run with H/h = 1.6 and frictional timescale 1 day. Also shown are their difference (thick, dashed) and the integrated potential momentum in the interior (thick, solid). (B) As A, but for a different choice of the reference latitude y0 . Units are in m2 /s.
28
Potential Momentum
Zonal Wind
A
10
5B
40
5
35
30
−16 −24
z
z
−32
25 20
−40
10 15
1520
−48 −8 −56 −8 −56 −64
−5
−5
y
y
PV Gradient
5
2
1 M U
C
x 10
Surface M
D 0
1.5
−1 1 −2 0.5
∫ M dz
yy
M , U
yy
−3
0
−4 −5
−0.5
−6 −1 −7 −1.5 −2
−8 −9
y
y
Figure 4: As panels A, C, D in Fig. 2 and panel A in Fig. 3, but with frictional timescale 5 days.
29
0.44
1.6
1.7
B
A τ=∞
0.435
1.6
1.4
1.5
1.2
Eddy amplitude
Circulation
Growth rate
0.43 1.4
0.425
C
v’q’
1.3
q’2
τ=∞
1
0.8
v’2
0.42
0.415
0.41
0
50
τ (days)
100
1.2
0.6
1.1
0.4
1
0
50
τ (days)
100
0.2
0
50
τ (days)
100
Figure 5: As a function of the interior eddy damping timescale τ (A) Growth rate, (B) Net Circulation (c.f., Eq. 19) and (C) Eddy amplitude in the indicated norms. The solutions in panels B and C are normalized with the τ = ∞ solution.
30
No damping
15−day interior damping A
B
−0.5
−1
−1
z
z
−0.5
−1.5 −3
−1.5 −3 −3.5
−2.5
−2
−2.5 −2
y
y
5−day interior damping
15−day boundary damping
C
D
−0.5 −1
z
z
−0.5 −2 −1 −3.5
−2.5
−1.5
−1.5
−2
−2.5 −3
y
y
Figure 6: Interior eddy PV flux for the interior and boundary eddy damping timescales indicated. Units are in 10−5 s−1
31
