Dynamics II
Lecture: May 13, 2024 (Monday), 14:00 Prof. Dr. Gerrit Lohmann
Taylor-Proudman Theorem
\[ \mbox{Assume constant density } \rho_0 \mbox{ on a plane with constant rotation } f=f_0 \neq 0 \]
Taylor’s lab experiments: homogeneous fluid tends to move in vertical columns
Then,
\[ \frac{\partial v}{\partial z}=\frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0 \]
Flow is two-dimensional and does not vary in the vertical direction.
Theorem applies to slowly varying flows.
Physical origin: stiffness endowed to the fluid by rapid rotation of the Earth.
Taylor-Proudman Theorem: The calculation
\[ \mbox{Assume constant density } \rho_0 \mbox{ on a plane with constant rotation } f=f_0 \] \[ -f\;v = - \frac{1}{\rho_0} \frac{\partial p}{\partial x} \] \[ f\;u= -\frac{1}{\rho_0} \frac{\partial p}{\partial y} \] \[ g= -\frac{1}{\rho_0}\frac{\partial p}{\partial z} \] \[ \mbox{and the continuity equation is:} \quad 0=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z} \]
Taking z-derivative of the first \[ -f_0\frac{\partial v}{\partial z}=-\frac{1}{\rho_0}\frac{\partial}{\partial z}\left(\frac{\partial p}{\partial x}\right)=\frac{\partial}{\partial x}\left(-\frac{1}{\rho_0}\frac{\partial p}{\partial z}\right)=\frac{\partial g}{\partial x}=0 \] \[ \mbox{ Therefore for } f_0 \neq 0 \quad \frac{\partial v}{\partial z}=0, \frac{\partial u}{\partial z}=\frac{\partial v}{\partial z}=0 \] Flow is two-dimensional and does not vary in the vertical direction.
Vertical velocity & north-south currents
Taylor-Proudman theorem: flow cannot expand or contract in the vertical
Assumption that \(f=f_0\) can not be appropiate
\[ \quad \quad \quad \quad \frac{D}{Dt}\left(\zeta+f\right) + \left(\zeta + f\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0 \quad \]
\[ \mbox{poor man's vorticity:} \quad \boxed{\beta\;v \quad + \quad f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)= 0 } \quad \quad \quad \quad \]
Using the continuity equation, we obtain \[ \beta\;v = f \frac{\partial w_g}{\partial z} \] in the ocean’s interior, geostrophic flow.
Variation of Coriolis force with latitude allows vertical velocity gradients in the interior of the ocean, and the vertical velocity leads to north-south currents.
What drives the ocean currents?
Friction: transfer of momentum from atmosphere to oceanic Ekman layer
Vorticity dynamics for the ocean and include the wind stress term
\[ D_t u - f v = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{1}{\rho} \partial_z \tau_{xz} \] \[ D_t v + f u = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{1}{\rho} \partial_z \tau_{yz} \]
\[ \frac{D}{Dt} \left( {\zeta+f}\right) \, = \, \frac{1}{\rho} \underbrace{\left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right)}_{curl \, \partial_z \tau} \quad . \]
Sverdrup transport
\[ \beta v = \frac{1}{\rho } \, \mbox{curl} \, \partial_z \tau \, \]
\[ \int_{-H}^0 dz \, \beta v = \frac{1}{\rho } \, \int_{-H}^0 dz \, \mbox{curl} \, \partial_z \tau \, = \frac{1}{\rho } \, \mbox{curl} \, \tau \, \]
\[ V = \frac{1}{\rho \beta} \, \left( \frac{\partial \tau_{yz} }{\partial x} \, - \frac{\partial \tau_{xz}}{\partial y}\, \right) = \frac{1}{\rho \beta} \, \, \operatorname{curl} \, \tau \]
applied globally using the wind stress from Hellerman and Rosenstein (1983). Contour interval is \(10\) Sverdrups (Tomczak and Godfrey, 1994).
\[ V = \frac{1}{\rho \beta} \, \left( \frac{\partial \tau_{yz} }{\partial x} \, - \frac{\partial \tau_{xz}}{\partial y}\, \right) = \frac{1}{\rho \beta} \, \, \operatorname{curl} \, \tau \]
west: f increase => \(\zeta\) negative, friction: positive vorticity
east: f decrease => \(\zeta\) less negative, friction: positive vorticity
Low Pressure: Friction on the bottom matters
left: 50%
As the ground is approached from above, and the influence of friction becomes important, this balance changes. The friction slows down the primary circular motion, and thus the Coriolis force is lower and no longer sufficient for the fluid to flow parallel to the isobars.
There is a net inward force, and the air thus moves toward the center of the low.
The converse case is also interesting. Around a high-pressure system, the air rotation is anticyclonic to achieve balance. In this case for the air outside the bottom boundary layer, the inward Coriolis force balances the outward radial pressure gradient.
Ekman Pumping & Sverdrup Transport
The center of a subtropical gyre is a high pressure zone: clockwise on the Northern Hemisphere
Ekman surface currents towards the center of the gyre
The Ekman vertical velocity balanced by \[ w_E=w_g \] vertical geostrophic current in the interior
geostrophic flow towards the equator
returned flow towards the pole in western boundary currents
North Atlantic Current & Gulfstream
brings warm water northward where it cools.
returns southward as a cold, deep, western-boundary current.
Gulf Stream carries 40 Sv of 18°C water northward.
Of this, 15 Sv return southward in the deep western boundary current at a temperature of 2°C.
How much heat is transported northward ?
Calculation:
\[ \underbrace{ c_p}_{4.2 \cdot 10^3 Ws/(m^3 kg)} \, \cdot \, \underbrace{ \rho }_{10^3 kg/m^3 } \, \cdot \, \underbrace{\Phi}_{15 \cdot 10^6 m^3/s} \, \cdot \, \underbrace{\Delta T}_{(18-2) K } = 1 \cdot 10^{15} W \]
The flow carried by the conveyor belt loses 1 Petawatts (PW), close to estimates of Rintoul and Wunsch (1991)
The deep bottom water from the North Atlantic is mixed upward in other regions and ocean, and it makes its way back to the Gulf Stream and the North Atlantic. Thus most of the water that sinks in the North Atlantic must be replaced by water from the far South Atlantic and Pacific Ocean.
Ocean Conveyor Belt
Conveyor belt circulation
The the conveyor is driven by deepwater formation in the northern North Atlantic.
The conveyor belt metaphor necessarily simplifies the ocean system, it is of course not a full description of the deep ocean circulation.
Broecker’s concept provides a successful approach for global ocean circulation, although several features can be wrong like the missing Antarctic bottom water, the upwelling areas etc..
metaphor inspired new ideas of halting or reversing the ocean circulation and put it into a global climate context.
interpretation of Greenland ice core records indicating different climate states with different ocean modes of operation (like on and off states of a mechanical maschine).
Thermohaline ocean circulation
Modelled meridional overturning streamfunction in Sv 10^6 = m^3 /s in the Atlantic Ocean. Grey areas represent zonally integrated smoothed bathymetry
Estimates of overturning ?
It is observed that water sinks in to the deep ocean in polar regions of the Atlantic basin at a rate of 15 Sv. (Atlantic basin: 80,000,000 km^2 area * 4 km depth.)
– How long would it take to ‘fill up’ the Atlantic basin?
– Supposing that the local sinking is balanced by large-scale upwelling, estimate the strength of this upwelling.
Hint: Upwelling = area * w
– Compare this number with that of the Ekman pumping!
Estimates of overturning: Solution
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:
\[ area \cdot w = 15 \cdot 10^6 \, m^3 s^{-1}\]
\[ w = 0.1875 \cdot 10^{-6} m\;s^{-1} = 5.9 \cdot 10^{-15} m \; y^{-1}. \]
Ekman pumping \[ w_E \simeq 32 \, \, m \; y^{-1}. \]
Vorticity dynamics of meridional overturning (y,z)
\[ \frac{\partial}{\partial t} v \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial y} \quad - \quad f u \quad - \quad \kappa v \]
\[ \frac{\partial}{\partial t} w \quad = \quad - \frac{1}{\rho_0} \frac{\partial p}{\partial z} \quad - \quad \frac{g}{\rho_0} (\rho -\rho_0) \quad - \quad \kappa w \] \(\kappa\) as parameter for Rayleigh friction.
Using the continuity equation
\[
0 \quad = \quad \frac{\partial v}{\partial y} \quad +
\quad \frac{\partial w}{\partial z}
\]
\[ \mbox{ one can introduce a streamfunction } \Phi (y,z): v= \partial_z \Phi; w = - \partial_y \Phi \]
The associated vorticity equation in the (y,z)-plane is therefore
\[ \frac{\partial}{\partial t} \nabla^2 \Phi \, = \, - f \frac{\partial u}{\partial z} \quad + \quad \frac{g}{\rho_0} \frac{\partial \rho}{\partial y} \quad - \quad \kappa \nabla^2 \Phi \]
Amplitude of overturning
\[ \frac{d}{dt} \Phi_{max} \, = \, \frac{a}{\rho_0} (\rho_{north} - \rho_{south}) \, \, - \, \, \kappa \Phi_{max} \]
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.
=======================================================
Schematic picture of the hemispheric two box model (a) and of the interhemispheric box model
=======================================================
The Atlantic surface density is mainly related to temperature differences.
But the pole-to-pole differences are caused by salinity differences. }
Rotation and Hadley Cell
Box model
3) Boxmodel (Version using R-shiny)
How to setup and run the Boxmodel R-shiny app
password: DynamicsII2020
Box model
for the box model, you can download sevenbox_jupyter.ipynb
conda create -n jupyter-R
conda activate jupyter-R
conda install -y -c conda-forge pip notebook nb_conda_kernels jupyter_contrib_nbextensions
conda install -y -c conda-forge r r-irkernel r-ggplot2 r-dplyr
jupyter-notebook sevenbox_jupyter.ipynb
you have to download conda (or miniconda). Here a good web site: https://conda.io/projects/conda/en/latest/user-guide/install/index.html