I. N. PANAYOTOVA289

11

=,

eeeee e

y yzzzzzzy

il y

m

(23)

where means applying consequently the Fourier

and

sine transforms, respectively in the

- and

-dire-

ctions, while 1

indicates the inverse of this process.

Then the hontal winds can be approximated byrizo

these potentials with the order

2

0 1(0(1)

~=

ss s

uu u

(1)

yz

y

F

, (24)

.

.2. Horizontal Advection

he horizontal winds (24) are used to advect the surface

)

01(0)(1) (1)

~=

ss s

xzz

vv vG

2

T

potential temperature

in (7). The artificial dissi-

pation operator is giveny the eighth-order horizontal

hyper viscosity (4). This representation for dissipation

has little connection to the real physics in primitive

equations, and is used in numerical experiments to

control the buildup of variability on small grid-scales.

Applying Fourier and sine transforms consequently ()

to (4) gives the spectral form of the governing equatio

b

n

4

22

ˆˆ

sssˆ

=.

ss

uv kl

txy

(25)

This equation then is forced randomly at high wave-

le

. Numerical Simulations

o compare the properties of forced

ngths in the spectral space, and direct numerical simu-

lations are performed with a resolution of 512 × 256

horizontal wave numbers. Temporal discretization is rea-

lized by the second-order predictor-corrector finite-

difference scheme.

3

T1

QG

e ran bot

turbu-

lence with the freely evolving regime wh simu-

lations, with and without stochastic forcing. The forcing

term is introduced as a random field in the spectral space

supplied at high wavelength. Each of the simulations

starts from the same random initial conditions and the

time step is =0.01

in non-dimensional time units.

One non-dimensional time unit is approximately equal to

12 hour. The simulations are made for an area that has a

channel geometry with dimensions 10π in the zonal

and 6π in the meridional directions that correspond to

appromately 31,000 km length and 18,000 km width.

Note that non-dimensional length unit is about 1000 km.

The parameters were chosen to represent the mid-latitude

atmospheric dynamics

Rossby number =0

xi

.2

Meridional vorticity gradient =2

Vertical shear =0

wshear as particularly chosen zero in here the vertical w

order to avoid its influence. The evolution of the freely

evolving 1

QG

surface potential temperature from

random inions shown in Figure 1. As it was

initially found in [5] inclusion of

itial condit

-effect in the higher-

order nonlinear dynamics added ave term that com-

petes with inertia on large-scales and produced high me-

ridional asymmetries in the eddies deformation radius.

This novel feature was added to the standard for the

a w

-plane turbulence zonal asymmetry, i.e. formed zonal

s. The zonal jet formation is shown in Figure 2.

The evolution of the forced 1

jet

QG

turbulence

fo

ns of the formed

vo

(40,80) we ca-

lc

e of the zonal flow we calculated

tim

perty is in accordance with the well known from the ob-

rm random initial conditions is shgure 3. The

simulations in the forced regime exhibit not only ani-

sotropy in the eddies spatial and time scales, but also in

their orientation. In addition to the established wave-like

motion, after some time there is an evident tendency of

the flow to stretch vortices in preferred direction. As it

can be seen from the direct numerical simulations the

cold anomalies are always stretched in the north-

eastward direction, while the warm anomalies are stret-

ched in the north-westward direction.

However to catch exactly directio

own in Fi

rtices we used one point correlation method. The co-

rrelation function represents a statistical process, and as a

statistical quantity should be calculated over a long

interval of time. The correlation is large and positive if

the elements tend to be in a phase, i.e. positive picks tend

to occur together. The correlation is strongly negative, if

the elements are in the opposite phase, i.e. the peaks in

one occur when valleys are attained in the other. Finally,

the correlation function vanishes if the two variables are

90 degrees out-of-phase, i.e. one is passing through zero

at the peak or valley of the other.

For a fixed element with coordinates

ulated its correlation with the others elements, and

correlation function contour lines are shown in Figure 4.

The left-hand side panel of Figure 4 shows positive

correlation of the reference point (40,80) with the other

elements, while the right-hand side panel represents the

negative correlation with the reference point. Those dire-

ctions coincide with the observed north-western elon-

gation of the worm anomalies and the north-eastern elon-

gation of the cold anomalies in the the potential tem-

perature evolution (3).

To see the persistenc

e sequence of the zonal mean of the potential tem-

perature at each latitude and the contour map is given in

Figure 5. This map illustrates that zonal jets are indeed

formed and in addition there is a meridional (northern/

southern) meandering of the formed jets. This novel pro-

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