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Conservation of mass t M w If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove fluid – it would be… inflow rate – outflow rate

Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

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Page 1: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Conservation of mass

t

M w

If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove fluid – it would be…

inflow rate – outflow rate

Page 2: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Now consider a small cube, fixed in space, through which air is flowing

The x direction mass flux is given by 12 smkgu

Page 3: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Using a Taylor series expansion about the center point, the rate of inflow through side A would be:

zyx

ux

u

2

and through side B would be:

zyx

ux

u

2

As with the basin, the rate of accumulation of mass in the cube must be:

zyx

ux

uzyx

ux

ut

M x

22

zyxuxt

M x

Page 4: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Similarly:

zyxuxt

M x

zyxvyt

M y

zyxwzt

M z

The net mass accumulation in the cube would be:

zyxwz

vy

uxt

M

Dividing by the volume zyx

wz

vy

uxt

Page 5: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

wz

vy

uxt

This equation can be written as:

Vt

The flux form of the mass

continuity equation

Using the vector identity:

VVV

this equation can be written as:

0

VVt

or:0

1 V

dt

d

The velocity divergence form of the mass

continuity equation

Page 6: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Vt

The flux form of the mass

continuity equation

01

Vdt

d

The velocity divergence form of the mass

continuity equation

For the special case of an incompressible fluid, density doesn’t change following parcel motion so the continuity equation reduces to:

0 V The incompressible form

of the mass continuity equation

Finally, let’s derive the mass continuity equation in pressure coordinates…

Page 7: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

zyxV Consider a volume of air:

Using the hydrostatic equation gz

p

zg

p

or

We can write our volume as:g

pyxV

g

pyxVM

and the mass of the parcel as:

If we follow parcel motion, the mass of parcel should be conserved:

g

pyx

dt

dM

dt

d 0

Page 8: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

g

pyx

dt

dM

dt

d 0

Divide both sides by M

g

pyx

dt

d

pyx

gM

dt

d

M

01

Apply chain rule to RHS

0

1

yx

dt

pdpx

dt

ydpy

dt

xd

pyx

Recalling that

dt

pdv

dt

ydu

dt

xd,,

The equation can be reduced, in the limit that the volume approaches 0, to be

0

py

v

x

u The pressure coordinate form of the mass continuity equation

Page 9: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

0

py

v

x

u The pressure coordinate form of the mass continuity equation

Vt

The flux form of the mass

continuity equation

01

Vdt

d

The velocity divergence form of the mass

continuity equation

0 V The incompressible form

of the mass continuity equation

The various forms of the mass continuity equation so far

Page 10: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Finally, substituting the geostrophic and ageostrophic wind:

yfug

1

xfvg

1

0

py

v

x

u

0

py

vv

x

uu aggagg

011 22

py

v

x

u

yxfyxfagag

0

py

v

x

u agag

0

p

Vag

The ageostrophic wind form of the mass continuity equation

Page 11: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

0

py

v

x

u The pressure coordinate form of the mass continuity equation

Vt

The flux form of the mass

continuity equation

01

Vdt

d

The velocity divergence form of the mass

continuity equation

0 V The incompressible form

of the mass continuity equation

The various forms of the mass continuity equation

0

p

Vag

The ageostrophic wind form of the mass continuity equation

Page 12: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

The pressure coordinate form is particularly enlightening

0

py

v

x

u

We can integrate this equation between the top of a column and the surface

t

s

th

sh

pVV

V

h

stPhPh stVV

Page 13: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

stPhPh stVV

If there is no vertical air motion at the surface and there is convergence, air will rise, a direct outcome of mass continuity. Vertical motion and column divergence patterns are directly related.

Page 14: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Conservation of energy

We will consider mechanical energy, thermal energy, and total energy

Page 15: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

cos2sin2

1tan2

2

wvz

uK

x

p

a

uw

a

uv

dt

du

sin2

1tan2

22

uz

vK

y

p

a

vw

a

u

dt

dv

The complete momentum equations

Multiply (1) by u, (2) by v, and (3) by w to get energy equations

cos2122

ugz

p

a

vu

dt

dw

cos2sin2

tan

2

12

2222

uwuvz

uuK

x

pu

a

wu

a

vu

dt

du

sin2

tan

2

12

2222

uvz

vvK

y

pv

a

wv

a

vu

dt

dv

cos22

1 222

uwwgz

pw

a

wvwu

dt

dw

Page 16: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

cos2sin2

tan

2

12

2222

uwuvz

uuK

x

pu

a

wu

a

vu

dt

du

sin2

tan

2

12

2222

uvz

vvK

y

pv

a

wv

a

vu

dt

dv

cos22

1 222

uwwgz

pw

a

wvwu

dt

dw

Add equations together: note that earth curvature and Coriolis force terms all cancel!

hh Vz

KVgwpVwvu

dt

d 2

2222 1

2

Note that:dt

d

dt

dzggw

Move this to left side

hh Vz

KVpVwvu

dt

d 2

2222 1

2

Page 17: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

hh Vz

KVpVwvu

dt

d 2

2222 1

2

Kinetic Energy

Potential Energy

Work done by PFG Work done by Friction

A change in total mechanical energy of a parcel of air must come about by work done by the pressure gradient and frictional forces

Note that in geostrophic flow so the first term on the LHS is zero in geostrophic flow. Only the Ageostrophic wind component does work.

ptoparallelisV

Page 18: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

The thermodynamic energy equation

First law of thermodynamics can be expressed as:

dt

dp

dt

dTcQ v

dt

dp

dt

dTcQ p or

where is the diabatic heating rate and is the specific volumeQ

FVpVwvu

dt

dh

1

20

222simplifyingfriction notation

Mechanical energy equation

Total energy equation (add TEE and MEE)

FVpVwvu

dt

d

dt

dp

dt

dTcQ hv

1

2

222

Page 19: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

FVpVwvu

dt

d

dt

dp

dt

dTcQ hv

1

2

222

pVpV

1

pVt

p

dt

dp

Note that: and

So:

t

p

dt

dppV

1

Substituting: FVt

p

dt

dpwvu

dt

d

dt

dp

dt

dTcQ hv

2

222

FVt

pwvupTc

dt

dQ hv

2

222

The Energy Equation

Page 20: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

If flow is adiabatic , frictionless and

steady state , then:

0Q 0F

FVt

pwvupTc

dt

dQ hv

2

222

0t

p

.2

222

constwvu

pTcv

-For an atmosphere at rest, an increase in elevation results in a decrease in hydrostatic pressure

Special case of Bernoulli’s equation for incompressible flow

Page 21: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

.2

222

constwvu

pTcv

Other implications

In accelerating flow over a hill the pressure difference between p2 and p1 must be > hydrostatic

Page 22: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

Potential Temperature

Temperature a parcel of air would have if it were brought dryadiabatically to a pressure of 1000 mb. “Dry adiabatically” impliesNo exchange of mass or energy with the environment, and noCondensation or evaporation occurring within the air parcel.

Potential temperature equation derived from 1st law of thermodynamics:

pdRTdcdS dp lnln S = Entropycp = Specific Heat at constant pressureRd = dry air gas constant

For an adiabatic process, dS = 0

0lnln pdRTdc dp

Page 23: Conservation of mass If we imagine a volume of fluid in a basin, we can make a statement about the change in mass that might occur if we add or remove

0lnln pdRTdc dp

Integrate equation from an arbitrary temperature and pressure tothe potential temperature and a pressure of 1000 mb.

1000

lnlnP

Tp

d pdc

RTd

pc

R

T p

d 1000lnln

p

d

c

R

pT

1000