1218 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
Nonnormal Frontal Dynamics
YIZHAK FELIKS
Department of Mathematics, Israel Institute for Biological Research, Ness-Ziona, Israel
ELI TZIPERMAN AND BRIAN FARRELL
Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts
(Manuscript received 3 June 2009, in final form 14 October 2009)
ABSTRACT
The generalized stability of the secondary atmospheric circulation over strong SST fronts is studied using
a hydrostatic, Boussinesq, two-dimensional f-plane model. It is shown that even in a parameter regime in which
these circulations are stable to small perturbations, significant nonnormal growth of optimal initial pertur-
bations occurs. The maximum growth factor in perturbation total energy is 250 and is dominated by the po-
tential energy, which obtains a growth factor of 219 two to five hours after the beginning of the integration. This
domination of potential energy growth is consistent with the observation that the available potential energy
(APE) of the secondary circulation is larger by two orders of magnitude than the kinetic energy as well as with
the transfer of kinetic to potential perturbation energy at the beginning of the growth of the perturbations.
The norm kernel is found to significantly influence the structure of the optimal initial perturbation as well
as the energy obtained by the mature perturbations, but the physical mechanism of growth and the structure
of the mature perturbations are robust.
1. Introduction layer of cold air from the cool side of the SST front
toward the warm side, while aloft an opposite return
Strong SST fronts are found in several parts of the
flow is found. In cases with the synoptic wind blowing
oceans, usually in association with intense oceanic cur-
from the cold to warm side of the SST front, the sec-
rents such as the Gulf Stream in the Atlantic, the Kuroshio
ondary circulation is increased in the lower layer and
in the Pacific, and the Agulhas Current in the Indian
decreased in the upper layer, and vice versa when the
Ocean (O’Neill et al. 2005). Strong SST fronts are also
synoptic winds are in the opposite direction. The sec-
found in the upwelling regions off the western coast of
ondary flow in the lower level diverges and descends
California and in warm and cold core rings (Park et al.
over the cold side of the SST front and converges and
2006). The evolution of the atmospheric marine bound-
ascends over the warm side. The atmospheric stratifi-
ary layer (AMBL) in these regions has been studied ob-
cation at a height of 50–130 m over the warm water is
servationally and to some extent using analytical and
unstable due to the secondary flow that advects cold air
numerical models. Observational evidence shows that the
over the warm water (Sweet et al. 1981; Small et al.
SST front induces a secondary circulation in the AMBL
2008). The height of the AMBL over the warm side of
(see reviews by O’Neill et al. 2003; Small et al. 2008),
the SST front is greater by several hundreds meters than
resembling the sea breeze (Hsu 1984): the cool air over
the height over the cold water.
the cold water is denser than the warm air over the warm
The surface wind over the warm water is also about
water, resulting in flow at the lower level of the boundary
twice as strong as that over the cold water. This differ-
ence in the surface wind speed results in the observed
calmer surface on the cold water side. In the Gulf Stream
Corresponding author address: Yizhak Feliks, Dept. of Math-
there is a visible sharp boundary between the two sur-
ematics, Israel Institute for Biological Research, P.O. Box 19,
Ness-Ziona, 70450 Israel. faces (see Sweet et al. 1981, their Figs. 1 and 2; Chelton
E-mail: feliks@iibr.gov.il et al. 2006, their Fig. 5). This is also seen in spaceborne
DOI: 10.1175/2009JAS3214.1
 2010 American Meteorological Society
APRIL 2010 F E L I K S E T A L . 1219
synthetic aperture radar images (Weissman et al. 1980; wind speed, roughness of the sea surface, and cloud
Sikora et al. 1995, their Fig. 1). Similar surface roughness bands.
boundaries can be observed in other regions where The initial conditions leading to transient growth are
strong SST fronts are found, including the cold and referred to as optimal initial conditions (Farrell and
warm core rings of the Gulf Stream (Park et al. 2006) Ioannou 1996) and are obtained by solving an eigenvalue
and the Brazil–Malvinas confluence (Tokinaga et al. problem based on the linearized model equations. The
2005; O’Neill et al. 2003). In all of these studies the role of transient growth has been studied in the context
secondary circulation exists under weak to moderate of atmospheric cyclogenesis (Farrell 1988, 1989), atmo-
synoptic wind conditions. spheric predictability (Buizza 1995; Buizza and Palmer
The stronger surface wind seen over the warm side of 1995), the wind-driven oceanic circulation (Moore 1999),
the SST front is conventionally explained by two dif- the El Niño–Southern Oscillation variability (Moore and
ferent mechanisms. One is based on the unstable strat- Kleeman 1997; Penland and Sardeshmukh 1995), and the
ification found in the AMBL. This stratification allows thermohaline circulation (e.g., Lohmann and Schneider
increased vertical turbulence that transports momentum 1999; Tziperman and Ioannou 2002).
from the prevailing synoptic wind to the sea surface. This paper does not address the synoptic frontogen-
Small et al. (2008) suggest that the turbulent fluctuations esis and cyclogenesis problems. These problems have
of heat, moisture, and momentum observed over the been thoroughly studied, beginning with the studies of
warm side of the front may be transported deeper into Hoskins and Bretherton (1972) and Hoskins (1975). Fur-
the boundary layer by large eddies to increase the sur- thermore, nonnormal growth of large-scale waves on
face wind. An additional mechanism is suggested in these fronts was studied by Joly (1995).
which the secondary circulation accelerates the surface In the following sections we describe the model equa-
winds across the front, producing the stronger wind over tions (section 2), analyze the nonnormal dynamics (sec-
the warm part. A summary of the debate of the relative tion 3), and conclude (section 4).
contribution of these two mechanisms to the surface
wind can be found in Small et al. (2008). Unfortunately,
observational studies of the secondary circulation do not 2. Model equations
specifically discuss the variability about the mean sec-
ondary circulation, which is particularly relevant to the To simplify the analysis, we study a front in the (x, z)
goals of this paper. These studies typically filter the ob- plane, where x is the cross-frontal direction and z the
servations to eliminate rapid variability faster than about vertical direction, assumed homogeneous in the along-
five days. front direction. The perturbations are also assumed
Over the warm side of SST fronts the low-level cloud homogeneous in the along-front direction. This sim-
cover is greater, consistent with the ascent there, as ob- plification can be justified because the cross-frontal
served over the Gulf Stream and over the Agulhas Re- length scale is about 100 km, while the scale in the
turn Current (Small et al. 2008; Minobe et al. 2008; along-front dimension is larger by an order of magni-
Sublette and Young 1996, their Fig. 2). tude. The hydrostatic, Boussinesq, f-plane model equa-
Modeling studies of the dynamics of the mean sec- tions are
ondary circulation show features similar to the observed
circulation including an increase in the height of the ›u ›u ›u 1 ›p ›2 2  u › u1 u 1w f y 5 1K 1K ;
AMBL across the front, the wind over the warm side ›t ›x ›z r ›x h ›x2 y ›z2
m
of the front being significantly stronger than over the
›y ›y ›y ›2y ›2y
cold side, and the flow ascending over the warm SST 1 u 1w 1 fu 5K 1K ;
h 2 y 2
and descending over the cold SST (Warner et al. 1990; ›t ›x ›z ›x ›z
Giordani and Planton 2000; Doyle and Warner 1990; ›u ›w1 5 0;
Wai and Stage 1989; Huang and Raman 1988). ›x ›z
The purpose of this paper is to study the dynamics 1 ›p gr5 ;
of the variability of the secondary circulation. We use r ›z rm m
a model of perturbations to a frontal circulation ho- u r
mogeneous in the along-front direction to examine the 5 ;u r
m m
nonnormal growth of perturbations to a steady sec-
2 2
ondary circulation in the AMBL induced by a SST ›u ›u ›u › u › u1 u 1w 5K 1K .
h 2 y 2
front. We show that these perturbations can explain ›t ›x ›z ›x ›z
the above observations near SST fronts, including the (1)
1220 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
The wind components u, y, and w are prescribed to is the perturbation density, and similarly introduce small
vanish at the sea surface. At the upper boundary, u and perturbations to the other fields u9, y9, u9, etc. The
y are taken to be equal to a specified geostrophic wind boundary conditions for the perturbations are such that
ug and yg. The upper boundary, at 5440 m, is assumed u9, y9, w9, and u9 vanish at the upper and lower bound-
to be a rigid lid, with w 5 0 there. At the upper aries. At the horizontal boundaries, the normal de-
boundary u is held constant at 318.38 K and p is de- rivative of u9, y9, and u9 is set to zero, as in the fully
termined from the prescribed geostrophic wind. The nonlinear model.
prescribed sea surface temperature difference across The computational domain over which the nonlinear
the front is 58C and the front width is 100 km. Away model was integrated to find the base steady state is
from the front region the SST gradient vanishes. The 500 km wide in the x direction and extends from the
geostrophic circulation is assumed to be determined by surface to a height of 5440 m. The integration domain
larger-scale processes not represented here. On the for the linearized model used to analyze the nonnormal
short time scales of interest in this paper, the geo- dynamics extends over the part of the computational
strophic circulation is unaffected by the SST gradients, domain of the full nonlinear model extending between
which have a small spatial scale compared to a typical 100 and 450 km along the x axis and 0 to 3200 m along
tropospheric Rossby radius of deformation (Feliks et al. the z axis. While the domain does not include the entire
2004, 2007). free troposphere, we ran the nonlinear model with a
At the horizontal boundaries the normal derivatives height of 8000 m instead of 5440 m and the mean cir-
of u, y, and u are set to zero. Convective adjustment is culation was practically identical. We also repeated the
applied during the model integration when the modeled linear analysis of the first experiment with a height of
lapse rate is 1022 K km21, slightly larger than the neu- 2880 m instead of 3200 m, and the results did not
tral lapse rate. Numerical details and physical parame- change, indicating that the results are not sensitive to the
ters are summarized in the appendix. Note that these vertical extent of the model. Note that the plotted do-
equations include the Ekman dynamics expected to be main in the figures below is often smaller than the full
critical within the AMBL. model domain, as we often plot only parts of the domain
We run the nonlinear model forced by the SST front where interesting signals develop. See the appendix for
until the secondary circulation reaches a steady state. more details.
We linearized the model given in (1) about this steady Integrating the kinetic energy equation over the do-
secondary circulation. Then we let r5 rm 1 r9, where r9 main and utilizing the boundary conditions we find
›hE i  
k 1 › h g5 ( u92i1hy92i)5hu92u ihu9w9u i1 w9u9  hy9u9y ihy9w9y i K (h(u9)2i1h(y9)2i)
›t 2 ›t x z u x z h x x
 mx
e
 K (h(u9)2i1h(y9)2i 1)1 u9p91uu921uy92 ð2Þ
y z z r
m x
w
in which angle brackets denote the integral over the  	1›u
domain, square brackets the vertical integral with re- a5 g u ,m›z
spect to z, and braces the difference of the enclosed term
between the horizontal boundaries (i.e., the net flux into
the domain through the vertical side boundaries).
The integrated available potential energy equation is integrating over the domain, and using the boundary
obtained by multiplying the u9 equation by au9 with conditions
h i     * + * +› E
p 1 ›
2 2
5 hau92i  ›u g u9 ›a u9 ›a5 au9u9  w9u9 1 u 1 w K ha(u9)2i
›t 2 ›t ›x u 2 ›x 2 ›z h x
m
 K ha(u9)2

 
i1 [uau92 x] e . (3)
y z x
w
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FIG. 1. Vertical cross-front section [x (km), z (m)] of the steady secondary circulation as obtained by the nonlinear
model for u 5 3 m s21, y 5 0. The warm water is to the right of the front. (left) Potential temperature ug g with contours
in K; (right) Wind field vectors for u and w. The maximum in the cross-frontal circulation is max(u) 5 6.7 m s21 and for
the vertical wind max(w) 5 3.7 cm s21. Contours are of the along-front wind y (cm s21). Note that the vertical extent
of the plotted domain is smaller than the actual computational domain (see section 2 and the appendix).
The integrated total energy equation is therefore
h i          › E
T 1 › h 2i h ›u ›u ›y ›y ›u5 ( u9 1 y92i1hau92i)5 u92  u9w9  y9u9  y9w9  au9u9
›t 2 ›*t + * + ›x ›z ›x ›z ›x
u92 ›a u92 ›a
1 u 1 w K (h(u9)2i1h(y9)2i1ha(u9)2i)K (h(u9)2i1h(y9)2i1ha(u9)2i)
2 ›x 2 ›z h x x x y z z z 
1 xe
1 u9p91 uu92 1 uy91 uau92 . (4)
r
m x
w
Calculating the optimal initial conditions 3. Analysis
Using finite differencing the linearized equations for
a. Steady circulation
u9, y9, and u9 may be written in vector form as
P 1 5BP 5B
nP , (5) The steady circulation in the atmosphere in the lower
n 1 n 0
3200 m above the SST front, as obtained by integrating
where P is the state vector of anomalies on the model 2D the nonlinear model (1), with the geostrophic wind set
grid, P5 [u91, u92, . . . , u9k, y91, y92, . . . , y9k, u91, u92, . . . , u9k], k is the to ug 5 3 m s
21, yg 5 0, is shown in Fig. 1. The depth of
total number of grid points, and B is the 3k 3 3k propa- the AMBL is 320 m over the cold side of the SST front
gator (matrix) of the finite difference linearized model. and 1500 m over the warm side. Above the AMBL
Define the energy norm kernel, X, to be a matrix such there is a strong inversion over the cold side of the SST
that P(t)TXP(t) is the total perturbation energy at a front and a much weaker inversion over the warm side.
time t. The optimal initial conditions that maximize the The secondary circulation is prominent in the AMBL
state vector norm P(t)TXP(t) at a time t 5 n dt, subject and the layer above it. The structure of the steady cir-
to P(0)TYP(0) 5 1, where Y may be a different norm culation obtained in the nonlinear model—including
kernel (possibly constraining the amplitude of the op- the flow of cold air and return flow, ascending and de-
timal initial condition under a different norm), are found scending motions, and the along-frontal velocity—is
by solving the eigenproblem (Farrell 1988, 1989; Farrell consistent with observations of secondary circulations
and Ioannou 1996) as described in the introduction. Standing waves orig-
inating in the location of maximum speed and tilting
Y1BnTXBne5 le, (6) toward the warm side of the domain with wavelength
1500 m in the vertical and 120 km in the horizontal are
where e is the eigenvector. evident.
1222 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
FIG. 2. The maximum perturbation energy growth as a function FIG. 3. The energy growth of optimal initial perturbation at t 5
of optimizing time t: the total energy divided by 10, hET(t)i/ 12.1 h as a function of time: hET(t)i/hET(0)i/10 (solid line); the
hET(0)i/10 (solid line); the potential energy divided by 10, hEP(t)i/ potential energy, hEP(t)i/hET(0)i/10 (dashed line); and the kinetic
hET(0)i/10 (dashed line); and the kinetic energy, hEK(t)i/hET(0)i energy, hEK(t)i/hET(0)i (dotted line).
(dotted line).
maximum growth factor is 196, somewhat smaller than
A linear stability analysis of the propagator B in (5) the maximum growth (Fig. 2) that occurs for t 5 4.2 h.
shows that the absolute value of the largest eigenvalues In Figs. 4–6 the initial optimal perturbation and its evolved
of B is smaller than 1. The propagator is therefore stable state at t 5 6.4 and 12.1 h are shown. The initial per-
to small perturbations, and the growth in the energy of turbation is found to be concentrated over the cold side
initial perturbations can only arise from the nonnormality of the domain, mainly in the frontal region just above the
of the propagator. AMBL.
At later times the perturbation propagates toward
b. Nonnormal growth
the warm side owing to advection by the mean flow. In the
The norm kernels X and Y in (6) that we used in the first 6.4 h the perturbation reaches the warm edge of the
experiments of this section are both set to the total en- front and evolves into a well-organized u9, w9 cell (Fig. 5)
ergy norm ET based on (4). We calculate the initial in the AMBL with significant signals of potential tem-
conditions that lead to the maximal total energy at t 5 t, perature and along-front velocity. The horizontal scale
and the growth factors for the total, kinetic, and potential of the cell is about 50 km. At later times the evolution
energies that result at time t are defined as hET,k,p(t)i/ changes significantly, and the propagation speed of the
hET(0)i. The potential energy calculation takes into ac- perturbation toward the warm side decreases by a factor
count the spatial changes of the mean stratification via the of 2 (Fig. 6, t 5 12.1 h) due to the large decrease in the
variations of a(x, z). The growth factors at the optimi- mean horizontal velocity at the warm edge of the front
zation time t are shown in Fig. 2 for hETi, hEpi, and hEki (Fig. 1). The strengths of u9 and w9 in the cell, as well as
as a function of t; hETi and hEpi reach their maximum at of y9, increase significantly and an additional strong cell
t 5 4.2 h: the total energy growth factor is 249 and is with an opposite circulation develops in the cold side
nearly all due to potential energy growth. The kinetic of the main cell. The perturbations also expand to the
energy reaches its maximum growth at t 5 12.1 h, and upper layers above the AMBL. Between the two cells
the corresponding growth factor is 4.7. This large dif- a very strong updraft is observed. This strong updraft
ference in the time of the maximum growth of the po- would give rise to low-level clouds as observed in the Gulf
tential and kinetic energies is an indication of different Stream over the warm side of the front (Small et al. 2008;
dynamics in different stages of the evolution, as will be Minobe et al. 2008; Sublette and Young 1996). The u9
shown below. perturbations are advected into the upper layers and
To analyze the growth mechanism we examine in reach the top of the domain. At later times the pertur-
detail the evolution of the initial perturbation for an bations continue to propagate slowly toward the warm
optimizing time of t 5 12.1 h, the optimizing time for side of the front. At this time the amplitude of the per-
which hEki reaches its maximum growth. The energy turbations begins to decrease, as can be seen in Fig. 3 for
growth as function of time is shown in Fig. 3; hETi and Ek. At later times (t 5 17 h) the perturbations moves out
hEpi reach their maximum growth at t 5 6.4 h. The of the domain through the warm part of the front.
APRIL 2010 F E L I K S E T A L . 1223
FIG. 4. Optimal initial perturbation of expt 1 at op-
timizing time t 5 12.1 h over the front (showing only
the part of the domain over the front): (a) y velocity
(cm s21, contour interval 5 1.5); (b) potential temper-
ature (units are 5 0.01 K); (c) mean potential temper-
ature contours (units are 1 K) superimposed on wind
field vectors for the optimal perturbation of u9, w9:
max(u9) 5 5 cm s21; max(w9) 5 0.06 cm s21.
As is typical for nonnormal growth, the energy tendency 1/2hu92wa i*, and K ha(u9)2i*. These terms are larger
z y z
by at least one order of magnitude than the other terms
1 ›E (t)T 1 ›E (t) 1 ›E (t)5 P 1 K between t 5 0 and 15 h, shown in Fig. 8. The initial(7)
E (t) ›t E (t) ›t E (t) ›t
T T T positive perturbation u9 (Fig. 4) is elongated along the
line where a has the largest horizontal gradient. As this
changes with time, as shown for our case in Fig. 7. In u9 perturbation is advected toward the warm side of the
normal (exponential) growth the energy tendency and front and downward by the mean horizontal and vertical
the tendency terms are constant. The energy tendency flow u, w, it gains available potential energy. As the
has an upper bound that is a function of the mean field perturbations reach the warm side of the front, the hori-
only. This upper limit is the value of the energy tendency zontal gradients of u and a decrease, so do the above
in the limit t / 0. In our case the growth rate in this limit perturbation energy source terms. This, together with the
is 0.0025 s21, that is, four times larger than the maximum vertical diffusion, leads to the decrease of hEpi (Fig. 3).
energy tendency obtained in our case with t 5 12.1 h. In the evolution of kinetic energy we notice several
To understand the growth mechanism, we examine stages. During the first half hour, the largest term is the
the role of the different tendency terms in the energy buoyancy generation by hw9u9g/u0i* 521.4 3 1025 s21,
equations (2), (3), and (4) divided by hETi (these terms resulting in the transfer of kinetic to potential energy. At
are indicated by an asterisk). The main points are briefly later times this term reverses, increasing the kinetic
summarized in the following subsection. energy.
In the equation for hEpi* the dominant term during The Reynolds stress terms
the first 10 min is the conversion of kinetic energy to
potential energy, 2hw9u9g/u i* 5 1.4 3 1025 s210 . Later,  *  *  *
the significant terms are hau9u9uxi*, which we refer to  ›u  ›y  ›yu9w9 , y9w9 , y9u9
later as ‘‘the horizontal buoyancy flux,’’ 1/2hu92ua i*, ›z ›z ›xx
1224 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
FIG. 5. Optimal perturbation of expt 1 at t 5 6.4 h,
panels and contour levels as in Fig. 4: max(u9) 5
7.7 cm s21; max(w9) 5 0.12 cm s21.
are positive during most of the evolution, resulting in in- available potential energy increases rapidly owing to
creasing perturbation kinetic energy via energy extracted energy transfer from the mean thermal front via advec-
from the mean flow. tion by u9 and movement of the perturbations toward
However, this is not the case at all times and locations. a region with lower stability. The kinetic energy increases
For example, perturbation energy growth occurs due to due to conversion from potential energy and extraction
perturbation interaction with the front during the first from the mean shear, and there is momentum transfer
6 h, or so, until the perturbation reaches the warm edge between u9 and y9 by the Coriolis force.
of the front. After this point, we find that momentum As the perturbations are advected to the warm side of
(and energy) is transferred from the perturbation to the the front, the third stage of the evolution begins, ex-
mean flow, which tends to increase the mean surface tending over 7 , t, 17 h. During this stage the structure
flow (Figs. 1 and 5), with observational consequences to of u9 changes dramatically, and the potential energy de-
be discussed below. In addition, interaction between per- creases via upward advection of the perturbations by w
turbations and the mean flow produces effects in the free toward the inversion, over the warm edge of the front.
troposphere above the boundary layer, up to about 2 km, During the fourth stage, t . 17 h, the perturbations are
that play an important part in the growth process. located to the right of the front where the energy source
terms are very small, and the kinetic energy now de-
c. Summary of growth mechanism
creases because of the dissipation terms.
From the above analysis we conclude that the evolu- The above description of the evolution of the initial
tion of the perturbations may be divided into four main optimal perturbation for t 5 12 h is very similar to the
stages. The first stage is very short, lasting less than half an evolution of any optimal initial perturbations calcu-
hour, and during this stage the kinetic energy is converted lated for 2 , t , 13 h. The main differences between
to potential energy. In the second stage, ½ , t , 7 h, the the evolution of optimal perturbations calculated for
APRIL 2010 F E L I K S E T A L . 1225
FIG. 6. As in Fig. 5 but at t 5 12.1 h: max(u9) 5
15 cm s21; max(w9) 5 0.4 cm s21.
different values of t are in the timing of the maximum seem to explicitly address the difference in variability
growth, the time and length of each stage of the evolution, (e.g., rms wind variability) between the warm and cold
and the amplitude of the perturbations at the different sides of the front. Additionally, we find that the strong
stages. updraft that develops from the optimal initial perturbations
d. Observable consequences
In the above analysis we found that the perturbations
in u9, y9, and w9 grow mainly at the warm side of the
front. We also showed that these growing eddies result
in a momentum transfer from an altitude where the wind
is strong (above 300 m) down to the surface and can
contribute to the increasing of the mean surface wind
over the warm side of the SST front. Note that this
transfer is independent of the sign of the perturbation.
Our analysis has therefore two predictions that may be
checked against observations: first, that the wind vari-
ability, excited via the nonnormal transient amplifica-
tion, is expected to be stronger over the warm side of the
front and, second, that the mean wind should be stronger
over the warm side due to the momentum transfer by FIG. 7. Energy tendency of the optimal perturbation for optimiz-
ing time t 5 12.1 h as a function of time: [1/E (t)][›E (t)/›t] (solid
the growing perturbations. As mentioned in the intro- T T
line); the potential energy tendency, [1/E (t)][›E (t)/›t] (solid line,
T P
duction, the observations seem to suggest that the mean overlapping the total energy curve); and the kinetic energy tendency
wind is indeed stronger over this warm side, yet do not multiply by 100, 100 [1/ET (t)][›EK(t)/›t] (dashed line).
1226 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
FIG. 8. Tendency terms in the potential energy equation (3) for experiment 1: hu9u9(›u/›x)i
(solid line), hu92u(›a/›x)i (dotted line), hu92w(›a/›z)i (dash–dot line), Kyha(›u9/›z)
2i (dash–
dot–dot line), and 2hw9ug9/u0i (dashed line).
at the warm edge of the front penetrates into the in- of the maximally amplified perturbations is very simi-
version and reaches high altitudes. This can also explain lar, but the location of the maximum in this case is
the low-level clouds found in observations over the about 30 km farther toward the cold part of the front
warm SST side of fronts (e.g., over the Gulf Stream and relative to that in experiment 1. These differences be-
the Agulhas Return Current) (Small et al. 2008; Sublette tween the two experiments result from the differences
and Young 1996; Minobe et al. 2008). between their secondary circulations. The slower u re-
sults in slower advection of the perturbations toward the
e. Sensitivity to the prescribed large-scale
warm SST, so the perturbations have more time to ex-
geostrophic velocity
tract energy from the mean steady flow so that the
In this section we examine the influence of the pre- maximum growth of the kinetic energy is at t 5 15.6 h
scribed geostrophic wind on the secondary circulation (i.e., 3.5 h later than in experiment 1). The larger gra-
and the nonnormal growth. We found that, of all the dient in y tends to increase the rate of energy transfer
model variables, the two parameters denoting the geo- from the mean flow. On the other hand, the larger gra-
strophic wind components seem to result in the most dient of u in experiment 1 tends to increase the transfer
interesting and perhaps least expected sensitivity be- rate in experiment 1.
havior: therefore, we concentrate on these parameters. In experiment 3 we set the geostrophic wind to zero:
We refrain from showing the sensitivity to other pa- ug 5 yg 5 0. The steady secondary circulation for this
rameters to avoid extending the length of the presen- experiment, shown in Fig. 9, is substantially different
tation. In experiment 2 we decrease the geostrophic from the previous cases of experiments 1 (Fig. 1) and 2
wind to ug 5 1.5 and yg 5 0 m s
21. The resulting steady (not shown, similar to experiment 1). The differences in
circulation is very similar to that in experiment 1, Fig. 1. the wind component parallel to the thermal front, y,
The main differences are the larger gradient in y and between the steady state of experiment 3 and the pre-
slower u in this experiment. The maximum growth fac- vious runs have a large effect on the calculated optimal
tor of the total and potential energy is 205 (as compared perturbations. This is seen when comparing the optimal
with 249 in experiment 1 where ug 5 3 m s
21, yg 5 0) initial conditions for t 5 2.8 h in experiment 3 (Fig. 10)
and is obtained by t 5 2.8 h. The maximum growth with the optimal initial conditions for experiment 1 (Fig. 4,
factor of kinetic energy is 20 (compared to 4.7 in ex- note that this figure is for t 5 12.1 h, but the results are
periment 1) and is obtained by t 5 15.6 h. The optimal similar if t5 2.8 h is used). The evolved state from these
initial perturbation is similar to that of experiment 1 initial conditions for experiment 3, at t 5 2.8 h, is shown
(Fig. 4) in u9, w9, and y9, but less so in u9. The structure in Fig. 11. Since the perturbation does not move through
APRIL 2010 F E L I K S E T A L . 1227
FIG. 9. Vertical cross-front section of the steady secondary circulation as obtained by the nonlinear model in expt 3
for ug 5 0 m s
21 and yg 5 0, panels and contours as in Fig. 1: max(u) 5 5.4 m s
21; max(w) 5 12 cm s21.
the areas with largest shear, it does not extracts kinetic where g is a prescribed weight of the potential energy
energy from y, so the kinetic energy has a maximum relative to kinetic energy: g 5 1 corresponds to the true
growth factor of 0.84, one order of magnitude less than in total energy.
experiments 1 and 2. On the other hand, the growth of the We investigate two extreme experiments: experiment 4
potential energy is only 20% less than in experiment 1 with g 5 1023 and experiment 5 with g 5 103. In ex-
since the perturbation extracts potential energy from the periment 4 the optimal initial perturbation is constrained
mean thermal front and also propagates from a stable to to have only potential energy for any value of t, and we
less stable region. find that the structure of u9 is similar to the initial per-
In experiment 3b we set the geostrophic wind to ug 5 turbation in experiment 3 (the case where g 5 1). The
23 m s21, yg 5 0. The steady secondary circulation for maximum growth of the total and potential energy is
this experiment, shown in Fig. 12, is substantially differ- obtained for t 5 2.8 h and the growth factor reaches 98.
ent from the previous cases of experiments 1–3 (Fig. 1). The kinetic energy growth factor reaches its maximum
The opposite flow in the wind component normal to the value of 0.3 for t 5 7.7 h. The structure of the mature
thermal front, u, results in the advection of the pertur- perturbation is similar to the mature perturbation in
bation by the mean circulation from the warm and less experiment 3.
stable part of the front toward the cooler and more stable In experiment 5 the optimal initial perturbation is
part. This has a large effect on the maximum growth of constrained to have only kinetic energy and the struc-
the total energy factor, which is only 2.9 at t5 1.4 h, that ture of u9, y9, and w9 is found to be similar to the initial
is, two orders of magnitude smaller than in the previous perturbation in experiment 3. The maximum growth
runs. Also, the difference in the wind component par- factor of the total and potential energies is 122, obtained
allel to the front, y, between this experiment and ex- at t 5 2.8 h. The kinetic energy reaches its maximum
periments 1–2 results in a much weaker kinetic energy growth of 1.13 at t 5 0.7 h. In the initial perturbation u9
growth factor of 0.3. is located within the inversion just above the AMBL, so
the initial perturbation is later advected into the AMBL.
f. Kinetic versus potential energy growth explored
The structure of the mature perturbation is similar to the
using the norm kernel
mature perturbation of experiment 3.
In the above experiments we found that only a small From these two cases we can conclude that the po-
fraction of the available potential energy of the growing tential energy always has the largest and fastest growth.
anomaly is converted to kinetic energy. This is most We can identify two reasons for this observation. First,
prominent in the case in which ug 5 yg 5 0 since the the mean APE is larger by two orders of magnitude than
maximum growth in the kinetic energy is less than 1, the mean kinetic energy of the steady background flow.
while in the potential energy it is 213. In this section we Second, we showed above that during the first stage of
examine the sensitivity to different norm kernels in or- growth there is a significant transfer of kinetic to po-
der to obtain insight into this issue. Consider a norm tential energy, which seems to prevent the development
kernel that is of a modified energy form, of a meaningful kinetic energy growth in spite of the
significant mean shears that could be used to extract
Y 5 u92 1 y92 1 gau92, (8) perturbation kinetic energy.
1228 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
FIG. 10. Optimal initial perturbation in experiment 3
using an optimization time of t 5 2.8 h (showing only
the part of the domain over the front), panels and con-
tour levels as in Fig. 4: max(u9) 5 6 cm s21; max(w9) 5
0.15 cm s21.
4. Conclusions what we term a ‘‘horizontal buoyancy flux’’, which cor-
responds to hau9u9u i*. The finding that this term plays
x
Strong SST fronts are found in many parts of the an important role in the nonnormal growth is an im-
World Ocean including, for example, the Gulf Stream portant novel finding of this work since horizontal var-
front and the Kuroshio front in the Pacific Ocean. These iations in the mean state have not been considered in
SST fronts induce a significant atmospheric ‘‘secondary nonnormal growth analysis previously. Another source
circulation’’ across the front. In this paper we examined of growth of the available potential energy is the move-
the generalized stability of such steady secondary cir- ment of the perturbation from a region with stable strat-
culations to small perturbations. The secondary circu- ification to a region where the stratification is less stable.
lations examined are steady solutions to a nonlinear Kinetic energy extraction is also maximal in the front
numerical model of the dynamics in a vertical cross sec- area due to the large mean shear there. As the pertur-
tion perpendicular to the SST front that is assumed to be bation travels out of the frontal region, the role of these
homogeneous in the along-front direction. These solu- source terms becomes small and the growth of the po-
tions are stable to small perturbations, so the growth tential energy stops. We found the results to be espe-
of the energy of small perturbations is due to nonnormal cially sensitive to the prescribed synoptic geostrophic
dynamics. wind at the model’s top boundary and examined this
In the basic experiment (experiment 1) the maximum sensitivity in detail.
total energy growth factor and potential energy growth In all experiments done in this paper the available
factor are both ;250 and are obtained for an optimi- potential energy (APE) of the perturbation is two orders
zation time of t 5 4.3 h. The growth in the perturbation of magnitude larger than its kinetic energy. We identi-
potential energy is due to extraction of potential energy fied two reasons for this observation. First, the mean
from the steady thermal front in the boundary layer by APE is larger by two orders of magnitude than the mean
APRIL 2010 F E L I K S E T A L . 1229
FIG. 11. Perturbation of expt 3 at t 5 2.8 h (showing
only part of the domain over the front), panels and con-
tour levels as in Fig. 4: max(u9) 5 9 cm s21; max(w9) 5
0.37 cm s21.
kinetic energy of the steady background flow. Second, may only grow due yo nonnormal effects, as analyzed
there is significant transfer of kinetic to potential energy above. It would be of interest to compare the intrinsic
in the initial stages of the growth. variability of a low-viscosity nonlinear model with the
The steady state of our nonlinear model is stable with nonnormal growth found here, but this is beyond the
no intrinsic variability, consistent with the linear prop- scope of the present study.
agator being stable to small perturbations. Transient We emphasized several observable consequences of
eddies are therefore expected only if initial perturbations this study. Our results suggest that the growing pertur-
are explicitly added to the system. Such perturbations bations lead to a momentum transport from altitudes
FIG. 12. As in Fig. 9 but for expt 3b with ug 523 m s
21, yg5 0: min(u) 5 3.9 m s21; max(w) 5 3 cm s21.
1230 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 67
where the mean wind of the secondary circulation is approximated by centered differencing, and the diffusion
strong (above 300 m) down to the surface and can con- is solved explicitly. Vertical diffusion and advection are
tribute to the increasing of the mean surface wind, as solved implicitly to allow large time steps. Sponge layers
seen in the observations (Sweet et al. 1981; Chelton et al. are used in the five grid points adjacent to the horizontal
2006; Weissman et al. 1980). This mechanism may sup- boundaries. Within the sponge layers, the vertical dif-
plement the two mechanisms for the enhanced mean fusion coefficient is larger by an order of magnitude. The
wind over the warm part of the front, which were already time interval dt5 40 s. The physical parameters are Kh 5
proposed in the literature and summarized in the in- 5 3 107 cm2 s21, K 5 104 cm2 s21y , f 5 14.585 3 10
25
troduction. Note that our mechanism would be effective sin328, um 5 300 K, and g 5 981 cm s
22.
even if the synoptic wind is weak, in which case the The linearized model about the steady secondary cir-
previously proposed transfer of momentum from the culation was integrated in a partial domain with 70 grid
synoptic wind to the surface is expected to be less effi- points in the horizontal and 23 vertical levels at heights
cient. The cells resulting from the nonnormal growth 0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960,
would also enhance downward mixing of drier air from 1040, 1120, 1280, 1600, 1920, 2240, 2560, 2880, and
aloft, thereby drying the surface air over the warm part 3200 m. The time step in the linearized model is dt 5
of the front. Indeed, the relative humidity near the 10 s: all other numerical and physical parameters were
surface decreases over the warm water, as shown for the same as in the fully nonlinear model.
example over the Gulf Stream (Sweet et al. 1981, their
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