It is all about rotation

\[ \underline{D_t \Theta_d \quad = \frac{\partial \Psi_d}{\partial x_d} + \nabla_d^2 \Theta_d \quad} \]

Galerkin approximation: Get a low-order model

Scheme

\[ \mbox{ Saltzman (1962): Expand } \Psi, \Theta \mbox{ in double Fourier series in x and z: } \]

\[ \Psi (x,z,t) \, = \, \sum_{k=1}^\infty \sum_{l=1}^\infty \Psi_{k,l} (t) \, \, \sin \left(\frac{k \pi a}{H} x \right) \, \times \, \sin \left(\frac{ l \pi}{H} z \right) \] \[ \Theta (x,z,t) \, = \, \sum_{k=1}^\infty \sum_{l=1}^\infty \Theta_{k,l} (t) \cos \left(\frac{k \pi a}{H} x \right) \, \times \, \sin \left( \frac{l \pi}{H} z \right) \]

From partial differential equations to ordinary differential equations for \(\Psi_{k,l}(t)\) and \(\Theta_{k,l}(t)\).

Approximation: Just 3 Modes X(t), Y(t), Z(t)

\[ \frac{a}{1+a^2} \, \kappa \, \Psi = X \sqrt{2} \sin\left(\frac{\pi a}{H} x \right) \sin\left(\frac{\pi}{H} z \right) \]

\[ \pi \frac{R_a}{R_c} \frac{1}{\Delta T} \, \Theta = Y \sqrt{2} \cos\left(\frac{\pi a}{H} x\right) \sin\left(\frac{\pi}{H} z \right) - Z \sin\left(2 \frac{\pi}{H} z \right) \]

Rayleigh number Ra: Buoyancy & Viscosity

\[ \mbox{Motion develops if } \quad R_a = \frac{g \alpha H^3 \Delta T}{\nu \kappa} \quad \mbox{exceeds a critical } \quad R_c \]

As the Rayleigh number increases, the gravitational force becomes more dominant. The critical Rayleigh number can be obtained analytically for a number of different boundary conditions by doing a perturbation analysis on the linearized equations in the stable state.

\(R_c = \pi^4 \frac{\left(1+a^2\right)^3}{a^2} = \pi^4 \frac{27}{4}= 657.51\) with \(a^2 = 1/2\)

\[\mbox{When } R_a < R_c,\mbox{ heat transfer is due to conduction} \]

\[\mbox{When } R_a > R_c, \mbox{ heat transfer is due to convection.} \]

Lorenz system

Bifurcation at \[ r = R_a/R_c = 1\]

 
Famous low-order model:

\[ \dot X = -\sigma X + \sigma Y \]

\[ \dot Y = r X - Y - X Z \]

\[ \dot Z = -b Z + X Y \]

Lorenz system r=0.9

r=0.9
s=10
b=8/3
dt=0.01
x=1.1
y=0.1
z=11.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:100){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1}
plot(vx,type="l",xlab="time",ylab="x")
plot(vy,type="l",xlab="time",ylab="y")

Lorenz system r=24

r=24
s=10
b=8/3
dt=0.01
x=0.1
y=0.1
z=0.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:10000){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1}
plot(vx,vy,type="l",xlab="x",ylab="y")
plot(vy,vz,type="l",xlab="y",ylab="z")

North Atlantic Current & Gulfstream

Gulf Stream & North Atlantic Current

Part of deep ocean

brings warm water northward where it cools.

returns southward as a cold, deep, western-boundary current.

Ocean Conveyor Belt

Conveyor

Thermohaline ocean circulation

Overturning

Modelled meridional overturning streamfunction in Sv 10^6 = m^3 /s in the Atlantic Ocean. Grey areas represent zonally integrated smoothed bathymetry

=======================================================

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.

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

\[ \Phi = - c (\alpha \Delta T - \beta \Delta S) \]

Box Models

Schematic picture of the hemispheric two box model (a) and of the interhemispheric box model

=======================================================

THC3

1. The Atlantic surface density is mainly related to temperature differences.
   2. But the pole-to-pole differences are caused by salinity differences. }

Box model of the thermohaline circulation

Box model of the thermohaline circulation

Ocean: multiple equilibria in GCM

THC4

Ocean response: Hosing the North Atlantic

THC6

1. AMOC indices of North Atlantic hosing for different hosing areas. Units are Sv. Black line represents the unperturbed LGM experiment. Hosing is for the period 840–990. (b) Annual mean sea surface salinity anomaly between LGM and the perturbation experiment LGM with 0.2 Sv for the model years 900–950.

-> Multi-scale Ocean GCM

Conveyor belt: climate change

left: 55%

THC7

halting or reversing the ocean circulation

interpretation of Greenland ice core records

climate states with different ocean modes

Deglaciation

Abrupt climate change, termination of ice sheets, Climate System II

Ocean box model

THC

Ocean: Temperature

THC

      Tnln = Tnl + dts * ((Hfnl)/(rcz2)-(Tnl-Tml)*phi/Vnl);
      
      Tmln = Tml + dts * ((Hfml)/(rcz1)-(Tml-Tsl)*phi/Vml);
      
      Tsln = Tsl + dts * ((Hfsl)/(rcz2)-(Tsl-Td)*phi/Vsl);
      
      Tdn = Td + dts * (-(Td-Tnl)*(phi/Vd));

Atmosphere

THC

Box model

Box model with jupyter

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

Boxmodel with R-shiny

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

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: Vorticity from shear

Cape

Tomczak & Godfrey: Regional Oceanography

\[u=0, v=v\left(x\right)\]

\[\zeta= \partial v\left(x\right)/\partial x \]

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 \(\zeta\).

Cape

 

Example: Vorticity from shear

Cape

Tomczak & Godfrey: Regional Oceanography

\[u=0, v=v\left(x\right)\]

\[\zeta= \partial v\left(x\right)/\partial x \]

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 \(\zeta\).

Cape

Dietrich et al. (1980)

---

Cape

Steward, Oceanography

 

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

\[ \beta v = \frac{1}{\rho } \, \mbox{curl} \, \partial_z \tau \, \]

\[ \int_{-H}^0 dz \, \beta v = \frac{1}{\rho } \, \int_{-H}^0 dz \, \mbox{curl} \, \partial_z \tau \, = \frac{1}{\rho } \, \mbox{curl} \, \tau \, \]

\[ V = \frac{1}{\rho \beta} \, \left( \frac{\partial \tau_{yz} }{\partial x} \, - \frac{\partial \tau_{xz}}{\partial y}\, \right) = \frac{1}{\rho \beta} \, \, \operatorname{curl} \, \tau \]

Sverdrup

applied globally using the wind stress from Hellerman and Rosenstein (1983). Contour interval is \(10\) Sverdrups (Tomczak and Godfrey, 1994).

Ekman Pumping & Sverdrup Transport

Ekman

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:

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