showed that approximately 5% of the jet energy is spent to generate IW in a thin
thermocline (assuming the IW decay with the radial distance from the jet, this
estimate can be somewhat increased), while only 2–3% of the jet energy flux is
transported away from the jet by IW in the thick thermocline.
The difference in the energy fluxes can be explained as follows. The amplitudes
of jet oscillations are similar in the thin and thick thermoclines. However, the
temperature gradient in the thick thermocline is weaker and thus, the maximal
buoyancy frequency is lower. One of the spectral peaks of plume oscillations is
higher than the maximal buoyancy frequency and the waves at this frequency
cannot propagate. Moreover, numerical simulations by [23] revealed a horizontal
counterflow in the upper thermocline attributed to the turbulent entrainment by the
jet top in the thick thermocline. This counterflow results in the transfer of the part of
IW energy towards the plume, thus decreasing the total energy flux transferred
away from the jet by IW.
Another interesting feature that has been revealed by [23] is that the ratio of the
energy flux of IW to the jet at the inflow to the thermocline remains approximately
constant in the simulations in the thin thermocline and almost does not depend on
the control parameter, which in this case is the Froude number. This effect can be
explained considering simplified energy conservation similar to the estimates made
by [15] for fountains. Note that Burridge and Hunt considered a different type of
oscillations, namely the vortices appearing on the inclined surface formed by the
top of the fountain, and their propagation as interfacial waves. Here, we consider
internal waves generated by the vertical oscillations of a jet top as a whole.
A thin thermocline can be treated similar to the interface between two fluids of
different temperatures, so as a first approximation to the dispersion relation we take
w =
ffiffiffiffiffiffi
g 0 k
p
, where g′ is reduced gravity. Note that it can be deduced from the dispersion curves calculated in the presence of a horizontal current and without it [12]
that the dispersion curves for the first mode of IW at the generation frequency are
close to each other. Thus, an estimate for the group velocity is c gr ∼
ffiffiffiffiffiffi
g 0 λ
p
, where λ
is the wavelength. The estimates of the oscillation frequency from [20] is ω ∼ 0.4
U/D, where U is the jet velocity at the inflow to the thermocline and D is the jet
diameter. Thus, an estimate for the wavelength is λ ∼ g
′ D
2
̸ U
2 . The energy flux of
internal waves at distance R from the jet centre can be estimated as F iw ∼ c gr A
2 g
0 R
[37]. We obtain the following estimate for the ratio of IW energy flux to the jet
energy flux in the thermocline
E iw ∼ c gr A
2 g
0 R ̸ U
3 D
2
The amplitude of IW is proportional to the jet penetration height h, while
h ∼ Fr
2 [23]. Thus, one obtains from these simple estimates that E does not depend
on Fr which is consistent with the numerical results in [23].
Note, that the dependence of IW amplitudes A
2 on the outflow velocity U,
presented in [10] was approximated by a linear function similarly to the dependence
of the jet oscillations amplitude A
2 on the control parameter. However, the results
80
V. G. Bondur et al.
thermocline (assuming the IW decay with the radial distance from the jet, this
estimate can be somewhat increased), while only 2–3% of the jet energy flux is
transported away from the jet by IW in the thick thermocline.
The difference in the energy fluxes can be explained as follows. The amplitudes
of jet oscillations are similar in the thin and thick thermoclines. However, the
temperature gradient in the thick thermocline is weaker and thus, the maximal
buoyancy frequency is lower. One of the spectral peaks of plume oscillations is
higher than the maximal buoyancy frequency and the waves at this frequency
cannot propagate. Moreover, numerical simulations by [23] revealed a horizontal
counterflow in the upper thermocline attributed to the turbulent entrainment by the
jet top in the thick thermocline. This counterflow results in the transfer of the part of
IW energy towards the plume, thus decreasing the total energy flux transferred
away from the jet by IW.
Another interesting feature that has been revealed by [23] is that the ratio of the
energy flux of IW to the jet at the inflow to the thermocline remains approximately
constant in the simulations in the thin thermocline and almost does not depend on
the control parameter, which in this case is the Froude number. This effect can be
explained considering simplified energy conservation similar to the estimates made
by [15] for fountains. Note that Burridge and Hunt considered a different type of
oscillations, namely the vortices appearing on the inclined surface formed by the
top of the fountain, and their propagation as interfacial waves. Here, we consider
internal waves generated by the vertical oscillations of a jet top as a whole.
A thin thermocline can be treated similar to the interface between two fluids of
different temperatures, so as a first approximation to the dispersion relation we take
w =
ffiffiffiffiffiffi
g 0 k
p
, where g′ is reduced gravity. Note that it can be deduced from the dispersion curves calculated in the presence of a horizontal current and without it [12]
that the dispersion curves for the first mode of IW at the generation frequency are
close to each other. Thus, an estimate for the group velocity is c gr ∼
ffiffiffiffiffiffi
g 0 λ
p
, where λ
is the wavelength. The estimates of the oscillation frequency from [20] is ω ∼ 0.4
U/D, where U is the jet velocity at the inflow to the thermocline and D is the jet
diameter. Thus, an estimate for the wavelength is λ ∼ g
′ D
2
̸ U
2 . The energy flux of
internal waves at distance R from the jet centre can be estimated as F iw ∼ c gr A
2 g
0 R
[37]. We obtain the following estimate for the ratio of IW energy flux to the jet
energy flux in the thermocline
E iw ∼ c gr A
2 g
0 R ̸ U
3 D
2
The amplitude of IW is proportional to the jet penetration height h, while
h ∼ Fr
2 [23]. Thus, one obtains from these simple estimates that E does not depend
on Fr which is consistent with the numerical results in [23].
Note, that the dependence of IW amplitudes A
2 on the outflow velocity U,
presented in [10] was approximated by a linear function similarly to the dependence
of the jet oscillations amplitude A
2 on the control parameter. However, the results
80
V. G. Bondur et al.