Dynamics II
Dynamics II
- Thermohaline ocean circulation
- Vorticity dynamics of meridional overturning (y,z)
- North-south density gradient in an ocean basin
- Ocean: multiple equilibria in GCM
- Ocean: multiple equilibria in GCM
- Rossby number Ro
- Vorticity is the rotation of the fluid
- Example: Vorticity from shear
- Planetary and relative vorticity
- Examples for Vorticity: Ocean with depth h(x,y)
- Potential vorticity: Examples
- Taylor-Proudman Theorem
- Taylor-Proudman Theorem: The calculation
- Vertical velocity & north-south currents
- What drives the ocean currents?
- Sverdrup transport
- Ekman Pumping & Sverdrup Transport
- Ekman Pumping: vertical velocity at the bottom of the Ekman layer E
Lecture: May 11, 2026 (Monday), 14:00 Prof. Dr. Gerrit Lohmann
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
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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.
Vorticity equation
Eliminating pressure yields
\[ \frac{\partial}{\partial t} \nabla^2 \Phi = - f \frac{\partial u}{\partial z} + \frac{g}{\rho_0} \frac{\partial \rho}{\partial y} - \kappa \nabla^2 \Phi \]
The Coriolis term is typically absorbed into the damping term for simplicity.
Galerkin Approximation
We expand the streamfunction as
\[ \Phi(y,z,t) = \sum_{k=1}^{\infty}\sum_{l=1}^{\infty} \Phi_{k,l}(t) \sin\left(\frac{\pi k y}{L}\right) \sin\left(\frac{\pi l z}{H}\right) \]
Single-mode truncation
We retain only the dominant mode:
\[ \Phi(y,z,t) = \Phi_{max}(t) \sin\left(\frac{\pi y}{L}\right) \sin\left(\frac{\pi z}{H}\right) \]
This satisfies no-normal-flow boundary conditions.
\[ \nabla^2 \left[ \sin\left(\frac{\pi y}{L}\right) \, \sin\left(\frac{\pi z}{H}\right)\right] = - \left[ \left(\frac{\pi}{H}\right)^2 + \left(\frac{\pi}{L}\right)^2 \right] \, \sin\left(\frac{\pi y}{L}\right) \, \sin\left(\frac{\pi z}{H}\right) \] —
Projection onto the dominant mode
We project the vorticity equation onto the chosen basis function:
\[ \int_0^L dy \int_0^H dz \, (\cdot) \]
Time tendency term
\[ - \left[ \left(\frac{\pi}{H}\right)^2 + \left(\frac{\pi}{L}\right)^2 \right] \, \int_0^L dy \sin\left(\frac{\pi y}{L}\right) \, \int_0^H dz \, \sin\left(\frac{\pi z}{H}\right) \] With \[ \int_0^L dy \sin\left(\frac{\pi y}{L}\right) \, = \frac{L}{\pi} \int_0^{\pi} d\varphi \sin \varphi = \frac{L}{\pi} \cos \varphi |_0^{\pi} = \frac{2 L}{\pi} \] and \[ \int_0^H dz \, \sin\left(\frac{\pi z}{H}\right) \, = \frac{H}{\pi} \int_0^{\pi} d\varphi \, \sin \varphi = \frac{L}{\pi} \cos \varphi |_0^{\pi} = \frac{2 H}{\pi} \]
this yields \[ \int \frac{\partial}{\partial t} \nabla^2 \Phi = 4 L H \left(\frac{1}{L^2} + \frac{1}{H^2}\right) \frac{d \Phi_{max}}{dt} \]
Density forcing term
\[ \int \frac{g}{\rho_0} \frac{\partial \rho}{\partial y} = \frac{g}{\rho_0} H (\rho_{north} - \rho_{south}) \]
Friction term
\[ \int \kappa \nabla^2 \Phi = 4 L H \kappa \left(\frac{1}{L^2} + \frac{1}{H^2}\right) \Phi_{max} \]
Final low-order model
Combining all terms gives
\[ \frac{d}{dt} \Phi_{max} = \frac{a}{\rho_0} (\rho_{north} - \rho_{south}) - \kappa \Phi_{max} \]
with
\[ a = \frac{g L H^2}{4 (L^2 + H^2)} \]
Diagnostic limit
Assuming fast adjustment (e.g. Kelvin waves), we obtain
\[ \Phi_{max} = \frac{a}{\rho_0 \kappa} (\rho_{north} - \rho_{south}) \]
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.
North-south density gradient in an ocean basin
Primary north-south gradient in balance with an eastward geostrophic current: generates a secondary high & low pressure system, northward current
Connection to Box Models
The density difference can be written as
\[ \rho_{north} - \rho_{south} = - \rho_0 (\alpha \Delta T - \beta \Delta S) \]
leading to the classical Stommel-type relation
\[ \Phi = - c (\alpha \Delta T - \beta \Delta S) \]
Schematic picture of the hemispheric two box model (a) and of the interhemispheric box model
both hemispheres:
densities at high northern and southern latitudes are close,
the pole-to-pole differences are caused by salinity differences.
Ocean: multiple equilibria in GCM
Ocean: multiple equilibria in GCM
Rossby number Ro
\[ \underbrace{\frac{\partial \mathbf{v}}{\partial t}}_{ U/T \sim 10^{-8} } + \underbrace{\mathbf{v} \cdot \nabla \mathbf{v}}_{ U^2/L \sim 10^{-8} } = {\underbrace{- \frac{1}{\rho} \nabla p}_{ \bf \delta P/(\rho L) \sim 10^{-5}} + \underbrace{2 \mathbf{\Omega \times v}}_{ \bf f_0 U \sim 10^{-5} } + \underbrace{fric}_{ \nu U/H^2 \sim 10^{-13}}} \]
\[ Ro = \frac{ \mbox{Inertial (the left hand side)} }{ \mbox{Coriolis term } } \]
\[ Ro = \frac{(U^2/L)}{(f U)} = \frac{U}{f L} \quad \]
characterizes the importance of Coriolis acceleration
Ro is small when the flow is in a so-called geostrophic balance.
Vorticity is the rotation of the fluid
\[ \zeta \equiv \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \]
or in 3D:
\[ \equiv \nabla \times \boldsymbol{ u} \]
Example: Rigid body rotating
\[ \boldsymbol{ u} = \begin{pmatrix} u \\ v \\ w \end{pmatrix}\ = \boldsymbol{ \Omega} \times \boldsymbol{ r} = \begin{pmatrix} \omega_1 \\ \omega_2 \\ \omega_3 \end{pmatrix}\ \times \ \begin{pmatrix} x \\ y \\ z \end{pmatrix}\ = \begin{pmatrix} \omega_2 z- \omega_3 y\\ \omega_3 x- \omega_1 z \\ \omega_1 y - \omega_2 x \end{pmatrix}\ \]
Rotation vector
\[ \nabla \times \boldsymbol{ u} = \begin{pmatrix} \partial_x \\ \partial_y \\ \partial_z \end{pmatrix}\ \times \ \begin{pmatrix} u \\ v \\ w \end{pmatrix}\ = \begin{pmatrix} \partial_x \\ \partial_y \\ \partial_z \end{pmatrix}\ \times \ \begin{pmatrix} \omega_2 z- \omega_3 y \\ \omega_3 x- \omega_1 z \\ \omega_1 y - \omega_2 x \end{pmatrix}\ \]
\[ = \begin{pmatrix} \partial_y (\omega_1 y - \omega_2 x) - \partial_z (\omega_3 x- \omega_1 z ) \\ \\ \\ \end{pmatrix}\ \]
\[ = \begin{pmatrix} \omega_1 + \omega_1 \\ \omega_2 + \omega_2 \\ \omega_3 + \omega_3 \end{pmatrix}\ = 2 \boldsymbol{ \Omega} \]
Example: Vorticity from shear
Tomczak & Godfrey: Regional Oceanography
\[u=0, v=v\left(x\right)\]
\[\zeta= \partial v\left(x\right)/\partial x \]
Estimate for \(\zeta\) off Cape Hatteras:
the velocity changes by \(1 \, {m}{s}^{-1}\) in 100 km
\[ \zeta= \frac{\partial v}{\partial x} = \frac{ 1 \, {m}{s}^{-1}}{100 \, {km}} = 10^{-5} \, \frac{1}{s} \]
still much smaller than
\[ f= 2 \Omega \sin \varphi = 2 \, \frac{2 \pi}{day} \sin \varphi \approx \, 10^{-4} \, \frac{1}{s} \]
Planetary and relative vorticity
\[ \mbox{Absolute Vorticity }\equiv\left(\zeta+f\right) \]
\[ \frac{Du}{Dt}-f\;v = -\frac{1}{\rho}\frac{\partial p}{\partial x} \] \[ \frac{Dv}{Dt}+f\;u = -\frac{1}{\rho}\frac{\partial p}{\partial y} \]
\[ \mbox{subtract } \partial/\partial y \mbox{ of (u-equation) from } \partial /\partial x \mbox{ of (v-equation) } \]
Use \[ \frac{D}{Dt} f = v \, \partial_y f: \]
to obtain
\[ \underline{ \frac{D}{Dt}\left(\zeta+f\right) + \left(\zeta + f\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0 } \quad \]
Examples for Vorticity: Ocean with depth h(x,y)
\[ \mbox{Because of the continuity equation } \quad \partial_x \left( u h \right) + \partial_y \left( v h \right) \quad = \quad 0 \]
\[ \quad \frac{D}{Dt} h + h \left( \partial_x u + \partial_y v \right) = \quad 0 \]
Therefore, \[ \underline{ \frac{D}{Dt}\left(\zeta+f\right) + \left(\zeta + f\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0 } \quad \]
\[ \frac{D}{Dt}\left(\zeta +f\right)-\frac{\left(\zeta+f\right)}{h}\frac{Dh}{Dt}=0 \]
\[ \frac{1}{h} \frac{D}{Dt}\left(\zeta+f\right) - \left(\zeta + f\right) \frac{D_t h}{h^2} =0 \]
\[ \underline{ \frac{D}{Dt}\left( \frac{\zeta+f}{h}\right) = 0 } \quad \]
Potential vorticity is conserved along a fluid trajectory.
Potential vorticity: Examples
\[ \frac{D}{Dt}\left(\zeta+f\right) + \left(\zeta + f\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0 \quad \]
Ocean/Atmosphere with depth h(x,y)
\[ \frac{D}{Dt}\left( \frac{\zeta+f}{h}\right) = 0 \quad \]
Couples depth, vorticity, latitude
– Changes in the depth results in change in \(\zeta\).
– Changes in latitude require a corresponding change in \(\zeta\).
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
\[ D_t u - f v = - \frac{1}{\rho} \frac{\partial p}{\partial x} + \frac{1}{\rho} \partial_z \tau_{xz} \] \[ D_t v + f u = - \frac{1}{\rho} \frac{\partial p}{\partial y} + \frac{1}{\rho} \partial_z \tau_{yz} \]
\[ \frac{D}{Dt} \left( {\zeta+f}\right) - \frac{\left(\zeta+f \right)}{h} \frac{D}{Dt} h \, = \, \frac{1}{\rho} \underbrace{\left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right)}_{curl \, \partial_z \tau} \quad . \]
\[ \frac{D}{Dt} \left( \frac{\zeta+f}{h}\right) = \frac{1}{\rho \, h} \, \mbox{curl} \, \partial_z \tau \, \]
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).
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
Ekman Pumping: vertical velocity at the bottom of the Ekman layer E
\(w_E\) 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 \(w_E\) at the bottom 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:
\[ w_E \simeq - \frac{ \Delta \tau_{x}}{\rho \;f_0 \Delta y}\, \simeq \frac{1 }{10^3 kg m^{-3}} \frac{0.2 N m^{-2} }{10^{-4} s^{-1}\, \, 2 \cdot 10^6 m} \simeq 32 \, \, \frac{m}{yr} \]