Dynamics II: lecture 4

Lecture: May 3. (Monday), 14:00 Prof. Dr. Gerrit Lohmann

Tutorial: May 3. (Monday), ca. 15:30 Justus Contzen, Lars Ackermann

Time required for Sheet: 8 h

May 11, 14:00: Lecture (online G. Lohmann, 45 min)

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Fourth lecture (link)

Content in the script: Angular momentum and Hadley Cell, Engery balance model,Wind-driven ocean circulation

After the lecture: Read the script about Atmosphere and Ocean Dynamics (Chapter 4 and 5)

Reading/learning (the sections with a star are voluntary). It might take 80 min.

The Coriolis effect is one of the dominating forces for the large-scale dynamics.

\[ \underbrace{\frac{\partial \mathbf{v}}{\partial t}}_{ U/T \sim 10^{-8} } + \underbrace{\mathbf{v} \cdot \nabla \mathbf{v}}_{ U^2/L \sim 10^{-8} }= {\underbrace{- \frac{1}{\rho} \nabla p}_{ \bf \delta P/(\rho L) \sim 10^{-5} } + \underbrace{2 \mathbf{\Omega \times v}}_{ \bf f_0 U \sim 10^{-5} } + \underbrace{fric}_{ \nu U/H^2 \sim 10^{-13}}} \quad \]

complemented by the dynamics for the vertical component

\[ \underbrace{\frac{\partial w}{\partial t}}_{ \begin{smallmatrix} W/T \sim 10^{-11} \end{smallmatrix}} + \underbrace{\mathbf{v} \cdot \nabla w}_{ UW/L \sim 10^{-11} } - \underbrace{\frac{u^2 + v^2 }{a}}_{ U^2/a \sim 10^{-9} } = \underbrace{ - \frac{1}{\rho} \frac{\partial p}{\partial z}}_{ \bf P_0/(\rho H) \sim 10} + \underbrace{ g }_{ \bf \sim 10} + \underbrace{f^{(2)} u}_{ \bf \sim 10^{-5}} + \underbrace{\nu \partial_z^2 w}_{ \nu W/H^2 \sim 10^{-16}} \]

\[ \frac{\partial u}{\partial t}+ \mathbf{v} \cdot \nabla u - \, \underbrace{ \frac{u v \tan \varphi }{a} - \frac{u w }{a}}_{\mbox{metric terms} } = - \frac{1}{\rho} \frac{\partial p}{\partial x} + f v - \, \underbrace{ f^{(2)} w}_{\mbox{small}} + \nu \nabla^2 u \]

\[ \frac{\partial v}{\partial t}+ \mathbf{v} \cdot \nabla v - \, \underbrace{ \frac{u^2 \tan \varphi }{a} - \frac{v w }{a}}_{\mbox{metric terms} } = - \frac{1}{\rho} \frac{\partial p}{\partial y} - f u + \nu \nabla^2 v \]

Taylor-Proudman theorem: flow cannot expand or contract in the vertical

Assumption that \( f=f_0 \) can not be appropiate

\[ \quad \quad \quad \quad \frac{D}{Dt}\left(\zeta+f\right) + \left(\zeta + f\right)\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)=0 \quad \]

\[ \mbox{poor man's vorticity:} \quad \boxed{\beta\;v \quad + \quad f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)= 0 } \quad \quad \quad \quad \]

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.

Consider a zonally symmetric circulation (no longitudinal variations).

a) Show, for inviscid zonally symmetric flow, that \( {DA}/{Dt}=0 \) is consistent with the zonal component of the equation of motion \[ \frac{Du}{Dt} - f v = -\frac{1}{\rho}\frac{\partial p}{\partial x} \]

Consider a zonally symmetric circulation (no longitudinal variations).

a) Show, for inviscid zonally symmetric flow, that \( {DA}/{Dt}=0 \) is consistent with the zonal component of the equation of motion \[ \frac{Du}{Dt} - f v = -\frac{1}{\rho}\frac{\partial p}{\partial x} \]

b) Use \( DA/dt=0 \) to describe how the existence of the Hadley Circulation explains the subtropical jet in the upper troposphere. Value of u at 20°N?

Consider a zonally symmetric circulation (no longitudinal variations).

a) Show, for inviscid zonally symmetric flow, that \( {DA}/{Dt}=0 \) is consistent with the zonal component of the equation of motion \[ \frac{Du}{Dt} - f v = -\frac{1}{\rho}\frac{\partial p}{\partial x} \]

b) Use \( DA/dt=0 \) to describe how the existence of the Hadley Circulation explains the subtropical jet in the upper troposphere. Value of u at 20°N?

c) Describe near-surface trade winds: low-level flow influenced by surface friction

Consider a zonally symmetric circulation (no longitudinal variations).

a) Show, for inviscid zonally symmetric flow, that \( {DA}/{Dt}=0 \) is consistent with the zonal component of the equation of motion \[ \frac{Du}{Dt} - f v = -\frac{1}{\rho}\frac{\partial p}{\partial x} \]

b) Use \( DA/dt=0 \) to describe how the existence of the Hadley Circulation explains the subtropical jet in the upper troposphere. Value of u at 20°N?

c) Describe near-surface trade winds: low-level flow influenced by surface friction

d) Hadley Circulation in northern winter:

Circulation rises at 10°S with u=0, sinks at 20°N. Calculate u at 0°, 10°N, 20°N !

For inviscid axisymmetric flow, conservation of angular momentum implies \[ D_t (\Omega a^{2} \cos^{2} \varphi + u a \cos \varphi ) = 0 \]

Remember that \( y= a \varphi, dx = a \cos \varphi \, d \lambda \)

Here, we reformulate the planetary term: \[ D_t ( \Omega a^2 \cos^2 \varphi ) = v \partial_y ( \Omega a^2 \cos^2 \varphi ) = \Omega a v \partial_\varphi ( \cos^{2} \varphi ) \] \[ = -2 \Omega a v \sin \varphi \cos \varphi = -f v \cdot a \cos \varphi \]

For inviscid axisymmetric flow, conservation of angular momentum implies \[ D_t (\Omega a^{2} \cos^{2} \varphi + u a \cos \varphi ) = 0 \]

Remember that \( y= a \varphi, dx = a \cos \varphi \, d \lambda \)

Here, we reformulate the planetary term: \[ D_t ( \Omega a^2 \cos^2 \varphi ) = v \partial_y ( \Omega a^2 \cos^2 \varphi ) = \Omega a v \partial_\varphi ( \cos^{2} \varphi ) \] \[ = -2 \Omega a v \sin \varphi \cos \varphi = -f v \cdot a \cos \varphi \] Similar \[ D_t (u a \cos \varphi ) = a \cos \varphi D_t u + u \cdot v \, \, \partial_\varphi \cos \varphi \] where in the coordinate system \[ D_t u = (\partial_t + u \partial_x + v \partial_y) u \, + \, \frac{uv}{a} \tan \varphi \] (the last term is a metric term). Therefore and under the assumption \( \partial_x p = 0 \): \[ D_t u -fv =0 \]

flow is under the influence of surface friction: A will be progressively reduced:

\[ A_{low} < A_0 \]

\[ \Omega a^{2} \cos^{2}\varphi + u_{low} \, a \cos \varphi < \Omega a^2 \]

flow is under the influence of surface friction: A will be progressively reduced:

\[ A_{low} < A_0 \]

\[ \Omega a^{2} \cos^{2}\varphi + u_{low} \, a \cos \varphi < \Omega a^2 \]

\[ \mbox{Thus} \quad u_{low} < \Omega a \frac{\sin^2 \varphi}{\cos \varphi} \]

\[ \mbox{ and some } \varphi_0 \mbox{ north of the equator that } u_{low} \mbox{ becomes negative (eastward winds), } \]

the low level flow will be equatorward and eastward there.

(Note that \( \Omega a^2 = \frac{2 \pi}{86400 \, s } \cdot (6.371 \cdot 10^6 m)^2 = 3 \cdot 10^9 m^2 s^{-1}. \))

Consider the angular momentum \[ A = \Omega a^{2} \cos^{2} \varphi + u a \cos \varphi = A_0 \] with
\[ \quad A_0 = \Omega a^2 \cos^{2} (10^\circ S) = 2.952 \cdot 10^9 m^2 s^{-1} \]

Consider the angular momentum \[ A = \Omega a^{2} \cos^{2} \varphi + u a \cos \varphi = A_0 \] with
\[ \quad A_0 = \Omega a^2 \cos^{2} (10^\circ S) = 2.952 \cdot 10^9 m^2 s^{-1} \]

\[ \mbox{Therefore} \quad u = \frac{A_0 - \Omega a^2 \cos^{2} \varphi }{ a \cos \varphi } \]

\[ \mbox{at } \varphi =0^\circ, u = -13.9 \, m/s \] \[ \mbox{at } \varphi =10^\circ N, u = 0 \, m/s \] \[ \mbox{at } \varphi =20^\circ N, u = 42.8 \, m/s \]

in other lectures

in the script

may be in one exercise

Friction: transfer of momentum from atmosphere to oceanic Ekman layer

Vorticity dynamics for the ocean and include the wind stress term

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

Friction: transfer of momentum from atmosphere to oceanic Ekman layer

Vorticity dynamics for the ocean and include the wind stress term

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

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

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

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

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

Assumption: h is constant, and poor man's vorticity

\[ v \beta = \frac{1}{\rho} \, \left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right) \quad . \]

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

Assumption: h is constant, and poor man's vorticity

\[ v \beta = \frac{1}{\rho} \, \left( \frac{\partial}{\partial x} \, \partial_z \tau_{yz} - \frac{\partial}{\partial y}\, \partial_z \tau_{xz} \right) \quad . \]

Integrating over z, we receive \[ \beta \int_h^{0} dz \, v = \beta V = \frac{1}{\rho} \, \underbrace{\left( \frac{\partial \tau_{yz} }{\partial x} \, - \frac{\partial \tau_{xz}}{\partial y}\, \right)}_{\operatorname{curl} \, \tau} \]

The pressure terms are small because the Ekamn layer is thin (10-20 m)

The Ekman transports \( V_E, U_E \) describe the dynamics in the upper mixed layer: \[ f V_E = -{\tau_x/\rho} \ \] \[ f U_E = +{\tau_y/\rho} \] where \( U_E = \int_{-E}^0 u dz \) and \( V_E = \int_{-E}^0 v dz \)

depth-integrated velocities in the friction-dominated Ekman layer.

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

\( \operatorname{curl} \mathbf{\tau} \) produces a divergence of the Ekman transports leading to a vertical velocity \( w_E \) at the bottom of the Ekman layer.

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

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

\( \operatorname{curl} \mathbf{\tau} \) produces a divergence of the Ekman transports leading to a vertical velocity \( w_E \) at the bottom of the Ekman layer.

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

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:

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