64
N. Makarenko et al.
map (Fig. 2)). These symmetric waves have almost the same amplitude but one of
them seems to be extremely broad. Such a broadening occurs when the coefficient
D 1 from (18) nearly vanishes, so the higher-order nonlinearity becomes important.
This is the same effect as the balance of quadratic and cubic nonlinearities in the
weakly nonlinear KdV–mKdV—Gardner model [6, 7, 20]. Broadening of internal
waves was investigated theoretically by [8, 14, 23], and studied experimentally by
[5].
In addition to the other features, internal Froude numbers F 1 and F 2 also characterize the magnitude of velocity jump at the interface in the upstream flow. The
shear u 1 ≠ u 2 between the layers can trigger the develop of the Kelvin–Helmholtz
instability which provides non-stationary formation of billow trains [3, 21]. Constant
two-layer flow is linearly stable under long-wave perturbations if the inequality
|u 1 − u 2 | <
√
g(í µí¼ 1 − í µí¼ 2 )(í µí¼ 1 h 2 + í µí¼ 2 h 1 )
í µí¼ 1 í µí¼ 2
is true, and this flow is unstable in the opposite case. Exactly the same bound for
a variable difference |u 1 − u 2 | and variable layer thicknesses h 1 , h 2 follows from
the non-linear stability criteria predicted by the shallow water theory [4, 19]. As a
consequence, we have the stability domain
|
√
rF 1 − F 2 | <
√
1 + r
shown for r = 1 on the Fig. 2 as an inclined strip confined between shadowed triangles in the quarter-plane (F 1 , F 2 ). Figures 4 and 5 demonstrate fragments of quasisteady shear flow recorded on a 350 m mooring station located at a depth of 4720 m
at the entrance to the Romanche Fracture Zone [24]. Trains of internal waves modulated by tide propagate here along with the sharp temperature gradient near isotherm
0.85 ◦ C, which separates the lower layer of cold Antarctic Bottom Water (AABW)
from the overlying warmer water. Owing to this fact, the moored CTD/LADCP data
indicate permanently marginal stability of the flow with the Richardson number
0.25 < Ri < 1. Tidal amplification of the shear triggers the formation of small-scale
overturns which create long trains of the Kelvin–Helmholtz billows. Bold curves in
Figs. 4 and 5 show overlapped profiles of solitary waves calculated by the solution
(19). The solitary wave shown on Fig. 4 is relatively short, and the flow is apparently
non-symmetric due to intense breaking, which localizes the sharp wave-crest downstream. In contrast, Fig. 5 demonstrates a long series of weaker overturns, which are
distributed uniformly along with gently sloped wave top. It is interesting that similar
overturning near the middle part of the broad solitary wave was observed in laboratory experiments [5].
N. Makarenko et al.
map (Fig. 2)). These symmetric waves have almost the same amplitude but one of
them seems to be extremely broad. Such a broadening occurs when the coefficient
D 1 from (18) nearly vanishes, so the higher-order nonlinearity becomes important.
This is the same effect as the balance of quadratic and cubic nonlinearities in the
weakly nonlinear KdV–mKdV—Gardner model [6, 7, 20]. Broadening of internal
waves was investigated theoretically by [8, 14, 23], and studied experimentally by
[5].
In addition to the other features, internal Froude numbers F 1 and F 2 also characterize the magnitude of velocity jump at the interface in the upstream flow. The
shear u 1 ≠ u 2 between the layers can trigger the develop of the Kelvin–Helmholtz
instability which provides non-stationary formation of billow trains [3, 21]. Constant
two-layer flow is linearly stable under long-wave perturbations if the inequality
|u 1 − u 2 | <
√
g(í µí¼ 1 − í µí¼ 2 )(í µí¼ 1 h 2 + í µí¼ 2 h 1 )
í µí¼ 1 í µí¼ 2
is true, and this flow is unstable in the opposite case. Exactly the same bound for
a variable difference |u 1 − u 2 | and variable layer thicknesses h 1 , h 2 follows from
the non-linear stability criteria predicted by the shallow water theory [4, 19]. As a
consequence, we have the stability domain
|
√
rF 1 − F 2 | <
√
1 + r
shown for r = 1 on the Fig. 2 as an inclined strip confined between shadowed triangles in the quarter-plane (F 1 , F 2 ). Figures 4 and 5 demonstrate fragments of quasisteady shear flow recorded on a 350 m mooring station located at a depth of 4720 m
at the entrance to the Romanche Fracture Zone [24]. Trains of internal waves modulated by tide propagate here along with the sharp temperature gradient near isotherm
0.85 ◦ C, which separates the lower layer of cold Antarctic Bottom Water (AABW)
from the overlying warmer water. Owing to this fact, the moored CTD/LADCP data
indicate permanently marginal stability of the flow with the Richardson number
0.25 < Ri < 1. Tidal amplification of the shear triggers the formation of small-scale
overturns which create long trains of the Kelvin–Helmholtz billows. Bold curves in
Figs. 4 and 5 show overlapped profiles of solitary waves calculated by the solution
(19). The solitary wave shown on Fig. 4 is relatively short, and the flow is apparently
non-symmetric due to intense breaking, which localizes the sharp wave-crest downstream. In contrast, Fig. 5 demonstrates a long series of weaker overturns, which are
distributed uniformly along with gently sloped wave top. It is interesting that similar
overturning near the middle part of the broad solitary wave was observed in laboratory experiments [5].