NFYK22003U-Computational-Atmosphere-and-Ocean-Dynamics/Lecture7.html at main · team-ocean/NFYK22003U-Computational-Atmosphere-and-Ocean-Dynamics · GitHub

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <link rel="icon" href="favicon.png" type="image/x-icon">
  <link href="https://fonts.googleapis.com/css2?family=Noto+Sans+JP&display=swap" rel="stylesheet">
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/styles/default.min.css">
  <link rel="stylesheet" href="style.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.8.0/highlight.min.js"></script>
  <script>hljs.highlightAll();</script>
  <script>
    MathJax = {
      tex: {
        inlineMath: [['$', '$'], ['\\(', '\\)']],
        displayMath: [['$$', '$$'], ['\\[', '\\]']]
      },
      options: {
        renderActions: {
          addMenu: []
        }
      },
      svg: {
        fontCache: 'global',
        scale: 1.0
      }
    };
    document.addEventListener("DOMContentLoaded", () => {
    const sidebar = document.querySelector('.sidebar');
    const main = document.querySelector('.main-content');
    const wrapper = document.createElement('div');
    wrapper.className = 'layout';

    sidebar.parentNode.insertBefore(wrapper, sidebar);
    wrapper.appendChild(sidebar);
    wrapper.appendChild(main);
  });
  </script>
  <script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js" async></script>
</head>

<body>
  <div class="sidebar">
    <img src="Logo_NS-blackboard-Expand.png" alt="Sidebar Image">
    <nav>
      <ul class="nav-menu">
        <li><a href="index.html">Welcome to NFYK22003U</a></li>
           <ul class="sub-menu">
            <li><a href="CourseInformation.html">Course Information</a></li>
          </ul>
        <li><a href="Welcome.html">Scope and Sequence</a></li>
           <ul class="sub-menu">
            <li><a href="CoursePrerequisitesExam.html">Prerequisites Exam</a></li>
            <li><a href="IntroductiontoPythonProgramming.html">Introduction to Python Programming</a></li>
            <li><a href="MathsSpeedrun.html">Maths Speedrun</a></li>
          </ul>

        <li>
          <a href="Lecture1.html">Lecture 1 – Introduction and Transport Processes</a></li>
          <ul class="sub-menu">
            <li><a href="Project1.html">Project 1 – Oxygen Budget</a></li>
          </ul>
          <li><a href="Lecture2.html">Lecture 2 – The Dynamics of Rotating Planets</a></li>
          <ul class="sub-menu">
            <li><a href="Project2.html">Project 2 – The Sverdrup Relation</a></li>
          </ul>
          <li><a href="Lecture3.html">Lecture 3 – Origin of Carbon and Water on Earth</a></li>
          <li><a href="Lecture4.html">Lecture 4 – The Gulfstream</a></li>
          <ul class="sub-menu">
            <li><a href="Project3.html">Project 3 – Western Boundary Currents</a></li>
          </ul>
          <li><a href="Lecture5.html">Lecture 5 – Carbon and Plankton</a></li>
          <li><a href="Lecture6.html">Lecture 6 – Quasi-Geostrophy</a></li>
          <li><a href="Lecture7.html">Lecture 7 – Theory of the Ventilated Thermocline</a></li>
          <ul class="sub-menu">
            <li><a href="Project4.html">Project 4 – Shadow Zone</a></li>
          </ul>
          <li><a href="Lecture8.html">Lecture 8 – Waves</a></li>
          <li><a href="Lecture9.html">Lecture 9 – Atmospheric Thermodynamics</a></li>
          <ul class="sub-menu">
            <li><a href="Project5.html">Project 5 – Hadley Cell</a></li>
          </ul>
          <li><a href="Lecture10.html">Lecture 10 – Surface and Abyssal Circulation</a></li>
          <ul class="sub-menu">
            <li><a href="Project6.html">Project 6 – Overturning</a></li>
          </ul>
          <li><a href="Lecture11.html">Lecture 11 – Ocean Mixing and Overturning Circulation</a></li>
          <li><a href="Lecture12.html">Lecture 12 – Stability of the Atlantic Meridional Overturning Circulation</a></li>
          <ul class="sub-menu">
            <li><a href="Project7.html">Project 7 – Oscillations</a></li>
            <li><a href="TestExam.html">Test Exam</a></li>
          </ul>
          <li><a href="Lecture13.html">Lecture 13 – Applications of Machine Learning in CFD and GFD</a></li>
          <ul class="sub-menu">
            <li><a href="SailbyNightPhysics.html">Sail by Night Physics</a></li>
          </ul>
          <li><a href="Appendix.html">Appendix</a></li>

          <li><a href="References.html">References</a></li>
        </li>
      </ul>
    </nav>
  </div>

  <div class="main-content">
<h1>Lecture 7 – Theory of the Ventilated Thermocline</h1>


<img src="Figures/Lecture7_GasBubbleOceanSurface.png" alt="GasBubbleOceanSurface" style="margin-top: 1.5em; max-width: 100%; display: block;">
<div class="figure-caption">(Gas bubbles rise toward the ocean surface. Credit: MartinStr)</div>
<h2>Connecting Layer Depths with Velocities</h2>
Define reduced gravity:

$$
\gamma_1 = \frac{\rho_2 - \rho_1}{\rho_0} g, \quad \gamma_2 = \frac{\rho_3 - \rho_2}{\rho_0} g
$$
Geostrophic relation for layer 2:
$$
v_2 = -\frac{\gamma_2}{f} \frac{\partial z_3}{\partial x}
$$
Sverdrup balance in layer 2 from geostrophic relation and continuity:
$$
\beta \cdot \frac{\gamma_2}{f} z_3 = f w_e
$$

This uses the steady-state vorticity equation with $\beta v_2 = f w_e$, and substituting $v_2$ from the geostrophic relation.<br>
Differentiate with respect to $x$:
$$
\frac{\partial}{\partial x} z_3^2 = \frac{2f}{\beta \gamma_2} w_e
$$

Integrate to get $z_3$:

$$
z_3^2 = H_2^2 - \frac{2f}{\beta \gamma_2} \int_x^{x_e} w_e \, dx
$$

We integrate from a boundary $x_e$, assuming a boundary condition where $z_3 = H_2$ at $x = x_e$, meaning flat interface (no flow) at the boundary.
<br>
Assumption:
$$
H_2 = \text{constant} \quad \because u_2 = \frac{\gamma_2}{f} \frac{\partial z_3}{\partial y} = 0 \quad \text{at the boundary}
$$

So there's no meridional geostrophic flow at the boundary.


<h2>Ventilation</h2>
Ventilation connects the atmosphere and ocean interior through the transfer of fluid from the surface mixed layer into the ocean
interior. Water masses are formed in the surface mixed layer and acquire their characteristics through the exchange of heat, moisture
and dissolved gasses with the atmosphere. When the water masses are transferred beneath the mixed layer, they are shielded from
the atmosphere and only subsequently modify their properties by mixing in the ocean interior. Hence, the ventilation process helps
to determine the relatively long memory of the ocean interior, compared with the surface mixed layer.

<h2>Ocean Subduction</h2>
Start with the vertically integrated Sverdrup balance:

$$
\beta (v_1 h_1 + v_2 h_2) = f w_E
$$
Using $h_2 = -z_3$ and geostrophic balance $v_2 = \frac{\gamma_2}{f} \partial_x z_3$, we obtain:

$$
\gamma_2 \frac{\partial}{\partial x}(z_3^2) + \gamma_1 \frac{\partial}{\partial x}(z_2^2) = \frac{-2 f^2}{\beta} w_E
$$

This represents the 2-layer Sverdrup balance expressed in terms of layer interface heights.
Define the layer 2 potential vorticity:

$$
q_2 = \frac{f}{h_2} = \frac{f}{z_2 - z_3}
$$

Since $q_2 \sim$ constant along streamlines, and is set at the subduction point:

$$
q_2 = \frac{f_2}{-z_3}
\quad \Rightarrow \quad z_2 = \left(1 - \frac{f}{f_2} \right) z_3
$$

Substitute $z_2$ into the Sverdrup balance and solve for $h = -z_3$:

$$
h(x,y) = \left( \frac{D^2(x) + H_2^2}{1 + \frac{\gamma_1}{\gamma_2} \left(1 - \frac{f(y)}{f_2} \right)^2} \right)^{1/2}
$$

where:

$$
D^2 = \frac{-2f^2}{\beta \gamma_2} \int_x^{x_E} w_E dx
$$

This integral accumulates wind forcing from east to $x$ that shapes.<br>
At the wall (eastern boundary), flow must vanish. So streamline must bend southward, conserving potential vorticity:

$$
h = H_2 = \left( \frac{D^2 + H_2^2}{1 + \frac{\gamma_1}{\gamma_2} (1 - f/f_2)^2} \right)^{1/2}
\Rightarrow D^2 = H_2^2 \frac{\gamma_1}{\gamma_2} \left(1 - \frac{f}{f_2} \right)^2
$$
We can compute the western extent of the shadow zone from this $D(x)$!<br>
<br>

Ocean subduction involves the transfer of fluid from the mixed layer into the stratified thermocline. The upper ocean is
ventilated principally through the subduction process, while the deep ocean is ventilated through open-ocean convection and
cascading down coastal boundaries. The term “ocean subduction” makes the geologicial analogy of a subduction zone where a rigid
plate of the Earth’s lithosphere slides beneath a more buoyant plate and into the hotter part of the mantle.<br>


Subduction occurs throughout the global ocean: over recirculating wind-driven gyres within ocean basins (with horizontal scales
of several thousand kilometers), across frontal zones and convective, overturning chimneys (on horizontal scales of several hundred
to tens of kilometers). The reverse of the subduction process leads to the transfer of fluid from the main thermocline into the
seasonal boundary layer, which affects the properties of the mixed layer and air–sea interaction; this negative subduction is
sometimes referred to as obduction.<br>
<img src="Figures/Lecture7_OceanSubduction.png" alt="OceanSubduction" style="margin-top: 1.5em; max-width: 100%; display: block;">

<div class="figure-caption">(A schematic diagram showing isopycnals in the thermocline outcropping into a vertically homogeneous mixed layer. The subduction rate, $S$, measures the rate at which fluid passes into the stratified thermocline through the vertical and horizontal transfer from the mixed layer. In comparison, the wind-induced Ekman downwelling, $-w_e$, measures the vertical flux pumped down from the surface Ekman layer.
  Credit: Marshall, J.C., Nurser, A.J.G., and Williams, R.G. (1993).)</div>
Seasonal boundary layer is a region over which the mixed layer and seasonal thermocline occur. The base of the seasonal
boundary layer is defined by the maximum thickness of the winter mixed layer.<br>
The subduction process involves the seasonal cycle of the mixed layer. The mixed layer is vertically homogenous through the
action of convective overturning and turbulent mixing.
<ul>
  <li>At the end of winter, the mixed layer is at its maximum density and thickness, and overlies the main thermocline (where there is a strong vertical temperature gradient).</li>
  <li>During spring and summer, the surface warming forms a seasonal thermocline, which is capped by a thin, wind-stirred, mixed layer.</li>
  <li>During autumn and winter, the cooling of the surface ocean leads to a buoyancy loss and convective overturning. The mixed layer thickens and entrains fluid from the underlying thermocline until the cooling phase ceases at the end of winter.</li>
  <li>Fluid is subducted from the mixed layer into the thermocline during the warming in spring and summer.</li>
  <li>fluid is entrained into the mixed layer from the thermocline during the cooling in autumn and winter.</li>
</ul>
Reference: <a href="https://www.liverpool.ac.uk/~ric/lfs/pdf/encyc_2019_wm.pdf">Circulation | Ocean Subduction</a>


<img src="Figures/Lecture7_SeasonalOceanSubductionProcess.png" alt="SeasonalOceanSubductionProcess" style="margin-top: 1.5em; max-width: 100%; display: block;">
<div class="figure-caption">(A schematic diagram illustrating the seasonal cycle of the mixed layer following the movement of a water column.
   The mixed layer thins in spring and summer, and thickens again in autumn and winter. If there is an overall buoyancy input, the end of winter mixed layer becomes lighter and thinner from one year to the next.
  Consequently, fluid is subducted irreversibly from the mixed layer into the main or permanent thermocline. The mixed layer thickness is marked by the thick dashed line, and isopycnals $\rho_m$ outcropping at the end of winter into the mixed layer by the full lines, and the isopycnal identifying the base of the seasonal thermocline by the short-dashed line.
  The annual subduction rate, $S_{\text{ann}}$, determines the vertical spacing between the isopycnals subducted from the mixed layer in March for consecutive years 1 and 2 ($\tau_{\text{year}} =$ 1 year).
  Credit: Williams and Follows (2011).)</div>


<script>
  function copyToClipboard(btn) {
    const code = btn.nextElementSibling.innerText;
    navigator.clipboard.writeText(code).then(() => {
      btn.classList.add('copied');
      setTimeout(() => btn.classList.remove('copied'), 2000);
    });
  }
</script>


    <p><strong>Next</strong>: <a href="Lecture8.html">Lecture 8 – Waves</a></p>


<div class="footer">
  Qi-fan based on the course material by Markus Jochum<br>
  It is very much a work in progress! Have you spotted a mistake or an error on this page?
  Click <a href="mailto:qifan.wu@nbi.ku.dk">here</a> to tell me!<br>
  &copy; 2025 TeamOcean | NBI/KU
</div>
</div>

<script>
  function copyToClipboard(btn) {
    const code = btn.nextElementSibling.innerText;
    navigator.clipboard.writeText(code).then(() => {
      btn.classList.add('copied');
      setTimeout(() => btn.classList.remove('copied'), 2000);
    });
  }
</script>
</body>
</html>