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While geostrophic motion refers to the wind that would result from an exact balance between the
Coriolis force In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
and horizontal pressure-gradient forces, quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are ''almost'' in balance, but with
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
also having an effect.


Origin

Atmospheric and oceanographic flows take place over horizontal length scales which are very large compared to their vertical length scale, and so they can be described using the
shallow water equations The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
. The Rossby number is a dimensionless number which characterises the strength of inertia compared to the strength of the Coriolis force. The quasi-geostrophic equations are approximations to the shallow water equations in the limit of small Rossby number, so that inertial forces are an
order of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
smaller than the Coriolis and pressure forces. If the Rossby number is equal to zero then we recover geostrophic flow. The quasi-geostrophic equations were first formulated by
Jule Charney Jule Gregory Charney (January 1, 1917 – June 16, 1981) was an American meteorologist who played an important role in developing numerical weather prediction and increasing understanding of the general circulation of the atmosphere by devis ...
.


Derivation of the single-layer QG equations

In Cartesian coordinates, the components of the
geostrophic wind In atmospheric science, geostrophic flow () is the theoretical wind that would result from an exact balance between the Coriolis force and the pressure gradient force. This condition is called '' geostrophic equilibrium'' or ''geostrophic balan ...
are : = (1a) : = - (1b) where is the geopotential. The geostrophic vorticity : = can therefore be expressed in terms of the geopotential as : = (2) Equation (2) can be used to find from a known field . Alternatively, it can also be used to determine from a known distribution of by inverting the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
operator. The quasi-geostrophic vorticity equation can be obtained from the and components of the quasi-geostrophic momentum equation which can then be derived from the horizontal momentum equation : + f \hat \times \mathbf = - \nabla \Phi (3)
The material derivative in (3) is defined by : (4) :where is the pressure change following the motion. The horizontal velocity can be separated into a geostrophic and an ageostrophic part : (5)
Two important assumptions of the quasi-geostrophic approximation are :: 1. , or, more precisely \sim O(\text). :: 2. the beta-plane approximation with
The second assumption justifies letting the Coriolis parameter have a constant value in the geostrophic approximation and approximating its variation in the Coriolis force term by .Holton, J.R. (2004). Introduction to Dynamic Meteorology, 4th Edition. Elsevier., p. 149. However, because the acceleration following the motion, which is given in (1) as the difference between the Coriolis force and the pressure gradient force, depends on the departure of the actual wind from the geostrophic wind, it is not permissible to simply replace the velocity by its geostrophic velocity in the Coriolis term. The acceleration in (3) can then be rewritten as : = = (6)
The approximate horizontal momentum equation thus has the form : = (7)
Expressing equation (7) in terms of its components, : (8a) : (8b)
Taking , and noting that geostrophic wind is nondivergent (i.e., ), the vorticity equation is : (9)
Because depends only on (i.e., ) and that the divergence of the ageostrophic wind can be written in terms of based on the continuity equation :
equation (9) can therefore be written as : (10)


The same identity using the geopotential

Defining the geopotential tendency and noting that partial differentiation may be reversed, equation (10) can be rewritten in terms of as : (11)
The right-hand side of equation (11) depends on variables and . An analogous equation dependent on these two variables can be derived from the thermodynamic energy equation : (12)
where and is the potential temperature corresponding to the basic state temperature. In the midtroposphere, .
Multiplying (12) by and differentiating with respect to and using the definition of yields : (13)
If for simplicity were set to 0, eliminating in equations (11) and (13) yields : (14)
Equation (14) is often referred to as the ''geopotential tendency equation''. It relates the local geopotential tendency (term A) to the vorticity advection distribution (term B) and thickness advection (term C).


The same identity using the quasi-geostrophic potential vorticity

Using the chain rule of differentiation, term C can be written as : (15)
But based on the thermal wind relation, : .
In other words, is perpendicular to and the second term in equation (15) disappears. The first term can be combined with term B in equation (14) which, upon division by can be expressed in the form of a conservation equation Holton, J.R. (2004). Introduction to Dynamic Meteorology, 4th Edition. Elsevier., p. 160. : (16)
where is the quasi-geostrophic potential vorticity defined by : (17)
The three terms of equation (17) are, from left to right, the geostrophic ''relative'' vorticity, the ''planetary'' vorticity and the ''stretching'' vorticity.


Implications

As an air parcel moves about in the atmosphere, its relative, planetary and stretching vorticities may change but equation (17) shows that the sum of the three must be conserved following the geostrophic motion. Equation (17) can be used to find from a known field . Alternatively, it can also be used to predict the evolution of the geopotential field given an initial distribution of and suitable boundary conditions by using an inversion process. More importantly, the quasi-geostrophic system reduces the five-variable primitive equations to a one-equation system where all variables such as , and can be obtained from or height . Also, because and are both defined in terms of , the vorticity equation can be used to diagnose vertical motion provided that the fields of both and are known.


References

{{refend Fluid mechanics Synoptic meteorology and weather