Geostrophic adjustment

The geostrophic adjustment is a process in geophysical hydrodynamics in which the fields of an initial pressure and speed disturbance in a rotating fluid are mutually deformed until a steady equilibrium between the pressure gradient and the Coriolis force belonging to the steady velocity field has been established . This final state is called geostrophic equilibrium . This problem was first theoretically solved by Carl-Gustaf Rossby for two special cases and published in 1937 and 1938. The geostrophic adjustment is done by Poincaré waves that are radiated from the initial disturbance. The characteristic length scale on which the geostrophic equilibrium is established is called the Rossby radius . On rotating planets with an atmosphere as well as with oceans in which the corresponding Rossby radii are smaller than the radius of the respective planet, the geostrophic adjustment is a fundamental process that determines the dynamics of their atmospheres and oceans to a large extent.

Mathematical description

Geostrophic adjustments come from a variety of different initial combinations of flow and pressure distributions . However, two complementary cases are particularly clear. In the first case, a box-shaped jet stream is combined with a spatially constant pressure field (Rossby's adaptation problem). Alternatively, one can investigate the adaptation of a horizontal pressure gradient in a fluid at rest (complementary Rossby problem). This latter case is mainly dealt with in the relevant textbooks, see Gill (1982).

We consider the geostrophic adaptation using the example of an infinitely extensive, frictionless ocean with a flat bottom at depth z  = - H on the earth rotating with angular velocity , see z. B. Gill (1982). By neglecting the vertical accelerations, the swell is filtered out of the equations of motion for the liquid of the ocean. ${\displaystyle {\vec {\Omega }}}$

The vertically averaged equations for the horizontal velocity components of the hydrostatic fluid are

${\displaystyle {\frac {\partial u}{\partial t}}-fv=-g{\frac {\partial \eta }{\partial x}}}$,

${\displaystyle {\frac {\partial v}{\partial t}}+fu=-g{\frac {\partial \eta }{\partial y}}}$.

In the above equations are:

• t : the time
• x , y , z : the coordinates of a right-angled coordinate system with the zero point at sea level on the geographical latitude , z. B. positive to the east, positive to the north and positive directed against gravity. ${\displaystyle \phi }$
• u , v : the horizontal components of the velocity vector in the direction of the x and y axes.
• ${\displaystyle \eta }$: the deflection of the sea surface from its resting position.
• ${\displaystyle f=2{\vec {\Omega }}\sin \varphi }$, the Coriolis parameter .

For the continuity equation of the fluid considered incompressible, we get

${\displaystyle {\frac {\partial \eta }{\partial t}}+H\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}$,

To get an equation for the displacement of the sea ​​surface , the divergence of the horizontal components of the momentum is formed and the continuity equation is used

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-c^{2}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}\right)+fH\zeta =0}$,

where is the phase velocity of a long wave on the non-rotating earth and ${\displaystyle c^{2}=gH}$

${\displaystyle \zeta ={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}}$,

the vertical component of the rotation of the velocity field.

If f  = 0 is set in the equation for the displacement of the sea surface , one obtains an equation for one variable, namely the wave equation . In the case of a rotating liquid, the above equation suggests taking into account the change in rotation of the horizontal velocity field. For this purpose, we form the rotation of the momentum equations from which the equation for the time change of the vertical component of the rotation of the velocity, viz

${\displaystyle {\frac {\partial \zeta }{\partial t}}+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}$,

results. This means that the change in time on the rotating earth is equal to the negative divergence of the horizontal movement. If the continuity equation is used to eliminate the horizontal divergence, the result is ${\displaystyle {\frac {\zeta }{f}}}$

${\displaystyle {\frac {\partial }{\partial t}}\left({\frac {\zeta }{f}}-{\frac {\eta }{H}}\right)=0}$.

This equation is the linearized form of the equation for the conservation of the potential vorticity of a homogeneous liquid in a rotating coordinate system. It expresses that the potential vorticity retains its initial value at every point at all times. This result can be used to calculate the development over time of an initial disturbance of the sea surface.

In the following we consider a special form of an initial pressure disturbance, in which at time t = 0 the sea is at rest, namely u  =  v  = 0 and there is a stepped jump in the sea surface at x  = 0 parallel to the y- axis. The deflection of the sea surface is then given by

${\displaystyle \eta =-\eta _{0}\operatorname {sgn}(x)}$.

The potential vorticity is then for all time

${\displaystyle \left({\frac {\zeta }{f}}-{\frac {\eta }{H}}\right)={\frac {\eta _{0}}{H}}\operatorname {sgn}(x)}$.

If we insert the above relation into the wave equation for the deflection of the sea surface, we get

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-c^{2}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}\right)+f^{2}\eta =-f^{2}\eta _{0}\operatorname {sgn}(x)}$.

The stationary solution

In contrast to the non-rotating earth, a permanent deformation of the sea surface remains on the rotating earth from an initial disturbance of the sea surface. In the above case, the equation results for the stationary solution because of the independence from the y coordinate

${\displaystyle -c^{2}{\frac {d^{2}\eta }{dx^{2}}}+f^{2}\eta =-f^{2}\eta _{0}\operatorname {sgn}(x)}$.

The solution for , first given by Rossby (1938), is continuous and antisymmetric with respect to x and has the form ${\displaystyle \eta }$

${\displaystyle {\frac {\eta }{\eta _{0}}}=(\Theta (-x)-\Theta (x))\left(1-e^{-{\frac {|x|}{R}}}\right)}$.

Water level after the geostrophic adjustment of an initially calm ocean with a water level disturbance in the form of a step running parallel to the y-axis.

Here is the so-called Rossby radius of the deformation, within which the originally stepped disturbance of the water surface was deformed into a steady but permanent transition. It is ${\displaystyle R}$

${\displaystyle R={\frac {c}{|f|}}={\frac {\sqrt {gH}}{|f|}}}$.

The Rossby radius is the fundamental length scale for the behavior of liquids on the rotating earth under the restoring influence of gravitational force. All originally existing disturbances in the pressure field of a liquid on the rotating earth are deformed to such an extent that the permanent remaining pressure disturbances have length scales of the order of magnitude of the Rossby radius. The associated horizontal pressure gradients are in equilibrium with the Coriolis force of the stationary geostrophic currents.

${\displaystyle -fv=-g{\frac {\partial \eta }{\partial x}}}$,

${\displaystyle +fu=-g{\frac {\partial \eta }{\partial y}}}$.

For the case of an initially stepped pressure disturbance described above, we get the stationary geostrophic velocity

${\displaystyle v=-{\frac {g}{f}}{\frac {\eta _{0}}{R}}e^{-{\frac {|x|}{R}}}}$,

a current field focused around the original step within the Rossby radius.

Geostrophic flow after the geostrophic adjustment of an initially calm ocean with a water level disturbance in the form of a step running parallel to the y-axis.

With an initial water level and an acceleration due to gravity g  = 9.81 m / s ^ 2 and a baroclinic Rossby radius R  = 50 km of an ocean in subtropical latitudes (35 ° latitude), see Chelton et al. (1998), one obtains the maximum geostrophic velocity at the position of the initial step . This result agrees well with the maximum flow velocity observed in the Gulf Stream . ${\displaystyle \eta _{0}=1\;\mathrm {m} }$ ${\displaystyle v_{0}={\frac {g}{f}}{\frac {\eta _{0}}{R}}=2{,}3\,{\frac {\mathrm {m} }{\mathrm {s} }}}$

The volume transport T of the geostrophic flow is given by the integral over the area in the ( x , z ) -plane

${\displaystyle T=\int \limits _{-\infty }^{\infty }dx\int \limits _{-H}^{0}dz\;v(x,z)=-{\frac {2\eta _{0}gH}{f}}}$.

The transport no longer depends on the exact course of the water level, but only on the difference between the highest and lowest water level along the horizontal extent of the integration area and the water depth. Based on the values ​​used above and assuming a vertical expansion of the flow of H  = 1000 m, a transport of T  = 234 Sv (1 Sv = 1 Sverdrup = 1 million m ^ 3 / s) is obtained. This is an overestimation of the Gulf Stream's maximum transport by a factor of two, even though the maximum geostrophic flow was correctly estimated. The reason for this deviation of the geostrophic transport from the observation lies in the assumption of a vertical constant flow. In contrast, the observed flow velocities in the Gulf Stream decrease continuously from the surface to their maximum depth at approximately 1000 m due to the superposition of the barotropic and baroclinic flow components, which reduces the resulting volume transport.

The energy changes in geostrophic adjustment

If one looks first at the potential energy of the disturbance, then that of the initial disturbance per unit of length integrated along the x -axis is infinite. After the geostrophic adjustment, it is also infinitely large in contrast to the non-rotating earth. However, the change in potential energy between the initial pressure disturbance and the disturbance remaining after the geostrophic adjustment is finite. It amounts to

${\displaystyle \Delta P_{\mathrm {e} }=2{\frac {1}{2}}\rho g\eta _{0}^{2}\int \limits _{0}^{\infty }\left(1-\left(1-e^{-{\frac {x}{R}}}\right)^{2}\right)dx={\frac {3}{2}}\rho g\eta _{0}^{2}R}$

In the case of the non-rotating earth, all potential energy present in the initial disturbance is converted into kinetic energy . In the case of the rotating earth, however, only a finite proportion is converted into kinetic energy. The amount of kinetic energy per unit length in the y -direction in the steady geostrophic flow is

${\displaystyle K_{\mathrm {e} }=2{\frac {1}{2}}\rho Hg^{2}\eta _{0}^{2}(fR)^{-2}\int \limits _{0}^{\infty }e^{-2{\frac {x}{R}}}dx={\frac {1}{2}}\rho g\eta _{0}^{2}R}$

Surprisingly, only a third of the potential energy released is converted into the kinetic energy of the geostrophic equilibrium flow.

The other two thirds of the potential energy given off by the original pressure disturbance are emitted by the energy of dispersive Poincaré waves or waves of inertia, starting from the position of the original jump in the water level, in the positive and negative direction of the x- axis with the maximum group velocity , Cahn (1945). The geostrophic adjustment takes place behind the two wave fronts and Poincare waves with a group velocity close to zero, i.e. a period which is close to that of the inertial oscillation, remain behind. The characteristic time for the geostrophic adjustment is thus the time T _a , the wave front with the group velocity c needed to the removal of one Rossbyradius R set back. It follows ${\displaystyle c={\sqrt {gH}}}$

${\displaystyle T_{\mathrm {a} }={\frac {R}{c}}={\frac {1}{f}}}$.

This means that the geostrophic adjustment takes place within a period of inertia.

Meaning and occurrence

The Rossby adaptation problem explains why the motions in the atmosphere and ocean are almost always in geostrophic equilibrium and why Poincare waves are so common. Every force that causes a disturbance of an existing geostrophic equilibrium triggers the process of geostrophic adjustment with the emission of Poincare waves, which re-establishes a new geostrophic equilibrium within a period of inertia, in which the spatial variations of the pressure field are smoothed over the distance of a Rossby radius . The new state of equilibrium is not a state of rest, but that of a moving fluid, in which the Coriolis force and the pressure gradient force are balanced. The geostrophic flow is exactly divergence-free and therefore has no vertical component. The mechanical energy of the new state of equilibrium is less than the energy of the originally disturbed state. The energy difference is contained in the emitted Poincare waves, which have established the transition to the new equilibrium.

The results obtained above are only valid for frictionless fluids in reference systems with constant rotational speed. On earth, the atmosphere and the ocean outside of the turbulent boundary layers are almost frictionless. The frictional force causes a spiral movement out of the high pressure area and into the low pressure area. As a result, the pressure contrasts are balanced out within a time scale that depends on the ratio of the amounts of friction and Coriolis force. The greater friction in the turbulent boundary layer of the atmosphere over land areas results in a faster pressure equalization than over the ocean (e.g. hurricanes and low pressure areas in the oceans surrounding the Antarctic ).

On planets, the Coriolis parameter is latitude dependent. Then the above results apply with high accuracy if the Rossby radii are smaller than the radius of the planet. The Rossby radius of the atmosphere on earth is around 1000 km and the barotropic Rossby radius of the ocean is approximately 2000 km, Gill (1982). These Rossby radii are of the same order of magnitude as the Earth's radius . In this case the latitude dependence of the Coriolis parameter must be taken into account, which results in the occurrence of Rossby waves . Rossby waves radiate from all pressure patterns in rotating frames of reference with meridional variability of the Coriolis parameter, which have zonal gradients. The radiation of the Rossby waves from geostrophic pressure patterns leads to their dispersion over time scales that are large compared to the period of inertia. They 'flow' slowly, whereby the current is quasi-geostrophic. The baroclinic Rossby radii in the ocean are very small compared to the earth's radius. They are on the order of 10 to 100 km, Chelton et al. (1998). Geostrophic currents in the interior of the ocean with a length scale that is comparable to the baroclinic Rossby radius are particularly persistent in shape. This is particularly true of geostrophic eddies, which can have a lifespan of several years. Since pressure patterns with purely meridional pressure gradients do not emit Rossby waves, they are also very persistent.

literature

• Cahn, A., 1945. An investigation of the free oscillations of a simple current system. J. Meteorol. 2, 113-119
• Chelton, DB, RA deSzoeke, MG Schlax, K. El Naggar, and N. Siwertz, 1998. Geographical Variability of the First Baroclinic Rossby Radius of Deformation. J. Phys. Oceanogr., 28, 433-460.
• Gill, AE (1982). Atmosphere-Ocean Dynamics. Academic Press Inc. New York, London, Tokyo, ISBN 0-12-283520-4
• Rossby, CG (1937). On the mutual adjustment of pressure and velocity distributions in certain simple current systems. IJ Mar. Res. 1, 15-28
• Rossby, CG (1938). On the mutual adjustment of pressure and velocity distributions in certain simple current systems. II. J. Mar. Res. 2, 239-263

Web links

• Animation of Rossby's adaptation problem