Dynamics II: lecture 6

Lecture: May 31 (Monday), 14:00 Prof. Dr. Gerrit Lohmann

Tutorial: June 31 (Monday), ca. 15:30

Time required for Sheet 2: 8 h

Read Chapter 11 (The thermohaline circulation of the ocean) in Marchal and Plumb

Reading/learning might take 90 min.

6th lecture link

Content in the script: Deep ocean circulation, Conceptual models

Friction: transfer of momentum from atmosphere to oceanic Ekman layer

Vorticity dynamics for the ocean and include the wind stress term

\[ D_t u - f v = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{1}{\rho} \partial_z \tau_{xz} \] \[ D_t v + f u = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{1}{\rho} \partial_z \tau_{yz} \]

\[ \frac{D}{Dt} \left( {\zeta+f}\right) - \frac{\left(\zeta+f \right)}{h} \frac{D}{Dt} h \, = \, \frac{1}{\rho} \underbrace{\left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right)}_{curl \, \partial_z \tau} \quad . \]

\[ \frac{D}{Dt} \left( \frac{\zeta+f}{h}\right) = \frac{1}{\rho \, h} \, \mbox{curl} \, \partial_z \tau \, \]

\( w_E \) as the Ekman vertical velocity the bottom of the Ekman layer \[ w_E = - \int_{-E}^0 \frac{\partial w}{\partial z} dz = \frac{\partial}{\partial x} U_E + \frac{\partial}{\partial y} V_E \]

\( \operatorname{curl} \mathbf{\tau} \) produces a divergence of the Ekman transports leading to \( w_E \)

\[ w_E = \, \frac{\partial }{\partial x} \left( \frac{ \tau_{y}}{\rho \;f }\, \right) - \frac{\partial }{\partial y}\, \left( \frac{ \tau_{x}}{\rho \;f }\, \right) =\operatorname{curl}\left(\frac{\mathbf{\tau}}{\rho\;f}\right) \simeq \frac{1}{\rho\;f} \, \operatorname{curl} \mathbf{\tau} \]

The order of magnitude of the Ekman vertical velocity:

typical wind stress variation of \( 0.2 N m^{-2} \) per 2000 km in y-direction:

\[ w_E \simeq - \frac{ \Delta \tau_{x}}{\rho \;f_0 \Delta y}\, \simeq \frac{1 }{10^3 kg m^{-3}} \frac{0.2 N m^{-2} }{10^{-4} s^{-1}\, \, 2 \cdot 10^6 m} \simeq 32 \, \, \frac{m}{yr} \]

Modelled meridional overturning streamfunction in Sv 10⁶ = m³ /s in the Atlantic Ocean. Grey areas represent zonally integrated smoothed bathymetry

Timescale T to 'fill up' the Atlantic basin:

\[ T = \frac{ 80 \cdot 10^{12} \, m^2 \cdot 4000 \, m}{15 \cdot 10^6 \, m^3 s^{-1}} = 2.13 \cdot 10^{10} s = 676 \;years \]

Overturning is balanced by large-scale upwelling:

\[ area \cdot w = 15 \cdot 10^6 \, m^3 s^{-1} \]

\[ w = 0.1875 \cdot 10^{-6} m\;s^{-1} = 5.9 \cdot 10^{-15} m \; y^{-1}. \]

compare to Ekman pumping \[ w_E \simeq 32 \, \, m \; y^{-1}. \]

\[ \frac{\partial}{\partial t} v \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial y} \quad - \quad f u \quad - \quad \kappa v \]

\[ \frac{\partial}{\partial t} w \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial z} \quad - \quad \frac{g}{\rho_0} (\rho -\rho_0) \quad - \quad \kappa w \] \( \kappa \) as parameter for Rayleigh friction.

\[ \frac{d}{dt} \Phi_{max} \, = \, \frac{a}{\rho_0} (\rho_{north} - \rho_{south}) \, \, - \, \, \kappa \Phi_{max} \] with \[ a = g L H^2/4(L^2 + H^2) \, \]

This shows that the overturning circulation depends on the density differences on the right and left boxes.

It is simplified to a diagnostic relation

\[ \Phi_{max} = \frac{a}{\rho_0 \, \kappa} \, \, (\rho_{north} - \rho_{south}) \quad \]

because the adjustment of \( \Phi_{max} \) is quasi-instantaneous due to adjustment processes, e.g. Kelvin waves.

hemispheric (Stommel-type) or interhemispheric (Rooth-type)

\[ \Phi \, = \, - c \, ( \alpha \Delta T - \beta \Delta S ) \qquad , \] where \( \alpha \) and \( \beta \) are the thermal and haline expansion coefficients,

\( \Delta \) denotes the meridional difference operator applied to temperature T and salinity S, respectively.

single hemispheric view:

The meridional density differences are clearly dominated by temperature

Salinity difference breakes the temperature difference

both hemispheres:

densities at high northern and southern latitudes are close,

the pole-to-pole differences are caused by salinity differences.

Download link password: DynamicsII2020

1) go to the folder where the boxmodel is located (ui.R): setwd(' … ')

2) install.packages(“shiny”)

3) library(shiny)

4) runApp()

5) a browser window should open that displays the boxmodel controls

6) first click on the button Spin-Up to create a new spin up

7) a new window opens, click on Run SpinUp

8) afterwards click on Apply, then Cancel

9) define parameters for the model simulation

10) run the simulation by clicking on Start Simulation