1
2
A1. Calibration
The amplitude of the back-scattered signal is calibrated to volume backscattering
3
cross-section σb in units m−1 (a measure of the plume's ability to scatter sound
4
waves, which is dependent on the suspended-particle concentration and
5
turbulence-induced temperature fluctuations within the plume) using the following
6
sonar equation:
7
8
9
VSS = 20 ∗ log10(abs(datain )) − SL − D − G + TL − 20 ∗ log10(RS)
−10 ∗ log10(dV).
(1)
Within the equation, VSS = 10log10 σb is the volume scattering strength, which is
10
the logarithm of the volume backscattering cross-section; abs(datain ) is the
11
envelope of the raw back-scattered signals；SL is the source level of the transmitter；
12
D is the directivity of the transmitter; G is the combined gain at both the transmitter
13
and receiver ends；TL is the transmission loss caused by the absorption of sound by
14
seawater and spherical expansion；RS is the receiver sensitivity, which is used to
15
convert the units of the raw signal (datain ) from machine units to μPa; and dV is the
16
volume element insonified by sound waves. Among those parameters mentioned
17
above, SL and RS are measured through laboratory experiments; TL is calculated
18
using the Francois-Garrison model (Francois et al. 1982) with constant parameters:
19
depth = 2500 m, salinity = 35 psu, temperatue = 2 °C, and PH = 7.
20
21
A2. Doppler Velocity Measurement
22
We measure the component Vr (in the direction of the acoustic line-of-sight) of
23
plume velocity from the Doppler frequency shift fD (Hz) of acoustic backscatter
24
using the covariance method described in Jackson et al. 2003. The following equation
25
gives the relation between Vr and fD ,
Vr =
26
27
1
N
cfD
2fc
T
p
w
fD = 2π∆t angle [∑n=1
∫t=0 E(t)E ∙∗ (t + ∆t)dt ′ ]
(2)
(3)
28
In equation (2), c is the sound speed and fc ~ 400 kHz is the sonar operation
29
frequency in the Doppler mode. In equation (3), the operator angle[ ] calculates the
30
phase angle in radians of a complex number. E(t) is a demodulated complex signal
31
corresponding to a given azimuthal beam and a given ping, whose amplitude and
32
phase reflect the amplitude of the acoustic backscatter and its phase shift relative to
33
the transmitted pulses. The integral in equation (3) estimates the autocorrelation
34
function of E(t) at the time lag ∆t. Note that a rectangular window with length
35
Tw = 1 ms is used to truncate the received signal. A summation over NP = 40 pings
36
at each elevation angle reduces the uncertainty in the measurement caused by
37
turbulence and background noise. In addition, the standard deviation 𝑉𝑠𝑡𝑑 of 𝑉𝑟 is
38
calculated over the 40 pings, which is used as a measure of the statistical uncertainty
39
in the Doppler measurements (Jackson et al. 2003).
40
A3. Entrainment Coefficient Calculation
41
Tao et al. 2013 estimate the entrainment coefficient α of hydrothermal plumes
42
rising in stably stratified static environments using numerical simulations. Their
43
estimates are reported to be consistent with experimentally determined values but
44
somehow not explicitly given in their paper. We herein recalculate the entrainment
45
coefficient using the simulation results and method presented in Tao et al. 2103.
46
47
According to Morton et al. 1956, the maximum rising height of a buoyant plume
can be calculated as
1
Zm =
48
B 4
Ce (N03 ) ,
(4)
49
50
where B0 (m4 ⁄s 3 ) is the initial buoyancy flux of the plume and N (1/s) is the
51
buoyancy frequency, and
Ce = 1.148α−1/2
52
(5)
53
54
is a universal constant. By performing linear regression on the simulation results of
55
Zm with different combinations of B0 and N, Tao et al. 2013 obtain Ce = 4.05.
56
Substituting into equation (5) gives α = 0.083.
57
A4. Spectrum Calculation
58
The time series samples of vertical volume flux, centerline vertical velocity, and
59
expansion rate are not evenly spaced. The sampling is semi-regular with the sampling
60
interval oscillating around 3 hours with small amplitude. In addition, the time series
61
has gaps concentrating mostly in the sector before Oct 9th (see Figure 6 (a), (b) in the
62
main text). As result, a modified periodogram (Lomb-Scargle periodogram, Scargle
63
1982) is used for spectral analysis of the time series. The Lomb-Scargle periodogram
64
has the following mathematical form
65
66
PX (ω) =
N
1 [∑j Xj cosω(tj −τ)]
{
2
∑N
2
j cos ω(tj −τ)
2
+
[∑N
j Xj sinω(tj −τ)]
2
∑N
j sin ω(tj −τ)
2
},
where τ is defined by
67
N
tan(2ωτ) = (∑N
j sin2ωt j )⁄(∑j cos2ωt j ).
68
Within the above equation, ω = 2πf and f is the frequency of interest; X is the time
69
series data; N is the number of samples within the time series. The results shown in
70
Figure 7 of the main text are normalized to the variance of the time series (PX (w)⁄σ2X )
71
and are thus dimensionless.
72
73
The Lomb-Scargle periodogram has well defined statistics for unevenly spaced
74
samples. Assuming Xj are independent samples of normally distributed noise, the
75
significance level of a spectral peak Z = PX (w) (the chance of observing a spectral
76
peak greater than Z ) is (Scargle 1982).
Z
77
N
QZ = 1 − (∫ χ2k (z)dz) , k = 2,
0
78
where χ2k is the probability density function of the chi-square distribution with k
79
degrees of freedom. In order to smooth the spectrum, periodograms calculated at
80
different levels along the plume axis (S = 5 to 15 m at 0.5-m intervals) are averaged to
81
give the result shown in Figure 7. Since a total of 21 periodogram are used for
82
averaging, assuming the noise at different levels is independent, the significance level
83
of the smooth spectrum becomes
84
QZ = 1 − (∫0
N
MZ 2
χk (z)dz) ,
k = 2M, M = 21.
85
The dashed horizontal line in Figure 8 denotes the value of Z with the significance
86
level of 0.05(5%).
87
88
89
A5 Isolation of the Inertial Oscillations
We isolate the inertial oscillations from the mean plume centerline vertical
90
velocity component ⟨Wc ⟩ by fitting the following sinusoidal function to the time
91
series of ⟨Wc ⟩,
Wci = C + Acos(2πfc t) + Bsin(2πfc t)
92
1
t+T
⟨Wc (t)⟩dt is the mean value of the time series, fc = 1.5 cycle/day
93
where C =
94
is the inertial peak in the spectrum (Figure 7 in the main text), A and B are optimal
95
coefficients corresponding to the minimum of the mean square fitting error
∫
T t
R=
96
97
Wci (t))2 dt.
The amplitude of the inertial oscillations is
Amp = √A2 + B2 ,
98
99
1 t+T
∫ (Wc (t) −
T t
and the phase is
B
Pha = arctan (A).
100
101
A moving window of length T = 4 days is used to segment the time series to yield
102
the variation of the inertial oscillation amplitude as a function of time shown in Figure
103
9 in the main text.
104
Note that we use the least-square-fit method describe above instead of the
105
common band-pass filters (Thomson et al. [1990]) to extract the inertial oscillations
106
because the ⟨Wc ⟩ time series is unevenly sampled and has gaps. The least-square-fit
107
method is immediately applicable to unevenly sampled time series; whereas one has
108
to interpolate the ⟨Wc ⟩ time series onto an evenly spaced time axis before band-pass
109
filtering, and the interpolation process will introduce bias into the result.
110
111
112
A6 Quantifying the tidal loading effects
According to Morton et al. [1956], the centerline vertical velocity Wc of a
113
buoyant plume injecting into a uniform, static background fluid from a point source
114
(B0 ≠ 0, M0 =0, Q0 =0, where M0 = 1/2b2 W02 and Q0 = b2 W0 are the initial
115
momentum and volume fluxes) can be calculated as
1/3
B
Wc = q ( z0 )
116
,
(6)
λ2
B0 = 1+λ2 b2 W0 gαt ∆T.
117
(7)
118
Within the equations, B0 (m4 /s3 ) is the initial specific buoyancy flux of the plume;
119
z is the height above the source; q is an empirically determined constant, which is
120
assumed ~1; b, W0 and ∆T are the orifice radius, initial vertical flow rate and
121
temperature anomaly of the plume; αt is the thermal expansion coefficient; λ = 1.2
122
is the empirically determined ratio between the e-folding radii of the cross-sectional
123
distributions of temperature and vertical flow rate; g = 9.8 m/s2 is the gravitational
124
acceleration.
125
126
A plume from a finite source ( B0 ≠ 0, M0 ≠0, Q0 ≠ 0) is asymptotically
127
identical to the plume from a point source (B0 , 0, 0) at vertical distance Z0 below the
128
vent orifice. Therefore we rewrite equation (6) as
129
1/3
B
Wc = (z+Z0 )
(8)
0
130
before applying it to the hydrothermal plume observed in this study. Using the
131
formulations developed by Morton 1959 with minor modification, Z0 is calculated as
132
1/|γ|
Z0 = 3.253 − 3.126|γ|3/2 ∫−1 (t 5 + 1)−1/2 t 3 dt,
133
(9)
134
γ5 = 1 − Г,
135
Γ = 22 5α−1 (1 + λ )B0 V0−5 Q20 ,
9
2
1/2
136
where α = 0.08 is the entrainment coefficient and V0 = M0 . Substituting the
137
following parameters into equation (7) to (9) gives the mean values of Wc at 5, 10,
138
and 15 m above the orifice. Note that the parameters are chosen to match those
139
measured at Grotto. Increasing W0 and ∆T by 40% and 6% (see Section 4.2 of
140
the main text) and substituting into equations (7) to (9) along with the other
141
parameters gives the perturbed values of Wc at each height. Table A1 summarizes
142
the results. The ratios of deviations to means are also given in Table 2 in the main
143
text.
144
Z = 5, 10 and 15 m;
145
∆T = 350 °C;
146
W0 = 0.3 m/s;
147
b = 0.1 m
148
αt = 10−4 / °C
149
Table A1:
Z (m)
Mean (m/s)
Perturbed (m/s)
Deviation/Mean
5
0.0414
0.0462
11.6%
10
0.0355
0.0399
12.5%
15
0.0320
0.0361
12.9%
150
151
152
153
Figure A1
154
155
156
Figure A1. Histogram of the mean volume flux <Q> time series. The red dotted lines denotes the
cut-offs of the central 80% quantile.
157
Figure A2
158
159
Figure A2. Power Spectrum of a year-long seafloor pressure time series (April 1st, 2011 – Apri 1st,
160
161
162
163
2012) measured by a mooring (RCM-NE, NEPTUNE Canada) deployed approximately 3 km to the
north of Grotto within the axial valley. The dashed lines denote the 98% confidence interval of the
spectrum (solid line). Note that the spectrum has two spikes at semi-diurnal and diurnal
frequencies, and no spike within the inertial band.
164
References:
165
1. R.E. Francois and G.R. Garrison, Sound absorption based on ocean measurements:
166
Part I: Pure water and magnesium sulfate contributions, Journal of the Acoustical
167
Society of America, 72, 896-907, 1982.
168
2. Jackson, D. R., C. D. Jones, P. A. Rona, and K. G. Bemis, A Method for Doppler
169
acoustic measurement of black-smoker flow fields, Geochemistry Geophysics
170
Geosystems, 4, 1095-1107, doi:10.1029/2003GC000509, 2003.
171
3. Scargle, J. D., Studies in astronomical time series analysis. II. Statistical aspects of
172
spectral analysis of unevenly spaced data, The Astrophysical Journal, 263,
173
835-853, 1982.
174
4. Morton, B. R., Forced Plumes, Journal of Fluid Mechanics, 5, 151-163, 1959
175
5. Morton, B. R., G. Taylor, and J. Turner, Turbulent Gravitational Convection from
176
Maintained and Instantaneous Sources, Proceedings of the Royal Society of
177
London, 234, 1-23, 1956.