Divergance in the wind field.

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I'm wondering when does this approximation hold, and when does it not? For example, is there a certain size scale where you can generally use this?

eqntl5.jpg


rho is density
u-east west wind
v-north south wind
w-up down wind
 
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in near-barotropic environments that relationship holds, such as the tropics

In synoptic meteorology (baroclinic enviroments), the commonly used relationship is d[u(ag)]/dx + d[v(ag)]/dy + d[omega]/dP = 0 where ag stands for the ageostrophic component of the wind on isobaric surfaces and omega is the time tendency of pressure (represents vertical motion).

du/dx + dv/dy + d[omega]/dP = 0 also holds on isobaric surfaces since V = Vg + Vag and Vg is non-divergent


(PS: the above derivatives are all assumed to be partial, can't figure out how to insert the partial character into text)
 
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The exact equation and logic displayed in the opening statement holds in all situations. It's a form of the mass continuity equation. If divergence is 0 then from the formula 1/rho*d(rho)/dt + del(dot)U = 0, you know that d(rho)/dt must be 0.
 
The exact equation and logic displayed in the opening statement holds in all situations. It's a form of the mass continuity equation. If divergence is 0 then from the formula 1/rho*d(rho)/dt + del(dot)U = 0, you know that d(rho)/dt must be 0.

oh, i misunderstood the question. Yes, if the 3-dimensional divergence of a velocity field is zero than the time derivative of density must be zero.

However, the 3-d divergence (du/dx + dv/dy + dw/dz) usually isn't zero in baroclinic environments. This was my point earlier; the above can only be considered approximately zero in near barotropic situations.
 
I know that if the divergence of the wind field is zero it means density can't change. However, air is compressible so the divergence for air is not zero. There are instances though you can approximate air as being incompressible. What are the cases when you can do this?

edit:
Sorry just saw the post... if barotropic situations means that divergance is zero, what would be an example of when an environment could be considered barotropic? For example can you only do this for large scale models?
 
I know that if the divergence of the wind field is zero it means density can't change. However, air is compressible so the divergence for air is not zero. There are instances though you can approximate air as being incompressible. What are the cases when you can do this?

edit:
Sorry just saw the post... if barotropic situations means that divergance is zero, what would be an example of when an environment could be considered barotropic? For example can you only do this for large scale models?

a baratropic environment is one where density is a function of pressure only (independent of temperature). The result of this would be coincidence of isotherms/isopycnic (constant density) surfaces/isobars, and a handful of other "nice" mathematical simplifications. Unfortunately this situation is an idealization, and it never actually occurs in the atmosphere. Environments are usually closer to barotropic near the equator, so your approximation would hold fairly well there.

In mid-latitudes situations are usually pretty baroclinic (pressure is a function of both density and temperature). "Baroclinicity" can be measured the baroclinic vector, which is (ro^2)^-1 * [nabla ro X nabla P] where ro is density, X represents vector product. If the magnitude of the baroclinic vector is nearly zero, the environment is nearly barotropic and 3-d velocity divergence is nearly zero.
 
It is convenient in meteorology to use pressure as a vertical coordinate rather than geopotential height. In this case the continuity equation becomes as follows:

du/dx + dv/dy + d(omega)/dP = 0

where d( )/d( ) is a partial derivative
u and v are evaluated on isobaric surfaces
omega is defined as dP/dt and represents vertical motion

in quasi-geostrophic analysis, the horizontal wind vector is separated in to geostrophic and ageostrophic components: Vag + Vg = V

Sine the geostrophic component of the horizontal wind is non-divergent, the continuity equation becomes:

du(ag)/dx + dv(ag)/dy + d(omega)/dP = 0

where (ag) is the ageostrophic wind component
 
I'm wondering when does this approximation hold, and when does it not? For example, is there a certain size scale where you can generally use this?

eqntl5.jpg


rho is density
u-east west wind
v-north south wind
w-up down wind

This is the incompressible mass continuity equation. One has to be careful here, because the actual incompressibility condition is that the material derivative of density following the motion is zero. That is, density doesn't change following a given air parcel (i.e. d(rho)/dt = 0). The result of this is that the 3D divergence of the velocity field also is zero (the first part of your equation above: del dot V = 0.

The incompressibility condition does not hold in general for atmospheric flows, since air is compressible (it's density can change significantly due to pressure changes, especially at high speeds or over large vertical length scales, due to the stratification of the atmosphere). However, for small vertical length scales, such as over the depth of the boundary layer, and provided the wind speeds are significantly smaller than the speed of sound, the incompressibility assumption is actually a very accurate approximation and is used extensively in models of the boundary layer. In such a flow, density can still vary spatially, or change in time at a given fixed point.

The anelastic approximation is more often used when vertical length scales are large (such as over the depth of the entire troposphere), since density changes appreciably in the vertical and vertical motions can no longer be assumed to be incompressible. Instead of del dot V = 0, we write del dot (rhobar*V) =0, where rhobar is the "base state density" which is only a function of height. Thus, it is like a less restrictive form of incompressibility, where compressible motions are only allowed in the vertical. This is an excellent approximation for many atmospheric flows (even, especially if your purpose is qualitative understanding, for thunderstorms).

It is convenient in meteorology to use pressure as a vertical coordinate rather than geopotential height. In this case the continuity equation becomes as follows:

du/dx + dv/dy + d(omega)/dP = 0

where d( )/d( ) is a partial derivative
u and v are evaluated on isobaric surfaces
omega is defined as dP/dt and represents vertical motion

Correct, but this can be misleading in the context of the original question because the continuity equation in pressure coordinates as you have written is actually *not* incompressible. It is in fact the full continuity equation (d(rho)/dt = del dot V) re-written in pressure coordinates. The advantage to doing so is that density disappears explicitly from the equation, making it look a lot like the incompressible continuity equation, but it is still there, hiding :)
 
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