I44
HI( ' H A RI 1 I I. PESKIN
Because the energy in twodimensional turbulence cascades toward lower
wave numbers, it was essential to introduce a mechanism to remove energy
in order to obtain quasi-steady solutions. The energy removal mechanism
for this model was a "surface friction coefficient." that is. a vorticity sink
linear in vorticity and operative at zcro wave number. Physically, such a sink
might represent surface friction in the Earth's boundary layer. In addition to
energy removal, energy input is required, and this was acconiplished by
using a constant amplitude random phase forcing function. A normalized
random phase Fourier component (at wave numher 8) was generated using
the following equations:
(2.4)
F,+ I = R,F, + ( 1 - Rf)"*F,, I
where subscript n refers to discrete time. Equation (2.4) is an approximation
to
where
(2.6)
At is integration increment, and 7 is the correlation time.
f , was obtained by Gaussian selection of wave number amplitudes on a
square of width Zk, in k,, k,, space. This amplitude was controlled normalizitig components so that k.: + k, Z was a constant. Time correlation for the
forcing function was provided by a simple Markoff process as indicated by
R, in Eq. (2.6).
All dependent and independent variables were scaled by the amplitude of
the forcing function and the input wave number. The remaining parameters.
which could be varied. were nondimensional kinematic viscosity, correlation
time, surface friction coefficient, and mesh lengths. The last were fixed to a
64 x 64 grid and other results presented in this paper were for a given choice
of the parameters.
r/At = 1(1 + R&(l - R , )
2.1.2. Eirlrrian Field Results. Batchelor (1969) and Kraichnan (1967)
prcdictcd that energy spectrum for two-dimensional turbulence would
divide into two regions either side of the wave number for energy input. The
lower wave number portion of the spectrum would behave according to the
- 1 hw, and in this region energy cascades toward lower wave number. At
higher wave numbers the spectrum would behave according to the - 3 law,
and in this region enstrophy, that is mean square vorticity, cascades to the
higher wave numbers. In fact, there is some question about the correct
description of the higher wave number region and the validity of the - 3 law,
HI( ' H A RI 1 I I. PESKIN
Because the energy in twodimensional turbulence cascades toward lower
wave numbers, it was essential to introduce a mechanism to remove energy
in order to obtain quasi-steady solutions. The energy removal mechanism
for this model was a "surface friction coefficient." that is. a vorticity sink
linear in vorticity and operative at zcro wave number. Physically, such a sink
might represent surface friction in the Earth's boundary layer. In addition to
energy removal, energy input is required, and this was acconiplished by
using a constant amplitude random phase forcing function. A normalized
random phase Fourier component (at wave numher 8) was generated using
the following equations:
(2.4)
F,+ I = R,F, + ( 1 - Rf)"*F,, I
where subscript n refers to discrete time. Equation (2.4) is an approximation
to
where
(2.6)
At is integration increment, and 7 is the correlation time.
f , was obtained by Gaussian selection of wave number amplitudes on a
square of width Zk, in k,, k,, space. This amplitude was controlled normalizitig components so that k.: + k, Z was a constant. Time correlation for the
forcing function was provided by a simple Markoff process as indicated by
R, in Eq. (2.6).
All dependent and independent variables were scaled by the amplitude of
the forcing function and the input wave number. The remaining parameters.
which could be varied. were nondimensional kinematic viscosity, correlation
time, surface friction coefficient, and mesh lengths. The last were fixed to a
64 x 64 grid and other results presented in this paper were for a given choice
of the parameters.
r/At = 1(1 + R&(l - R , )
2.1.2. Eirlrrian Field Results. Batchelor (1969) and Kraichnan (1967)
prcdictcd that energy spectrum for two-dimensional turbulence would
divide into two regions either side of the wave number for energy input. The
lower wave number portion of the spectrum would behave according to the
- 1 hw, and in this region energy cascades toward lower wave number. At
higher wave numbers the spectrum would behave according to the - 3 law,
and in this region enstrophy, that is mean square vorticity, cascades to the
higher wave numbers. In fact, there is some question about the correct
description of the higher wave number region and the validity of the - 3 law,