Cumulus

• Forced; pushed upward by external forces, i.e. lifting by surface convergence, topography etc.

• Active; growing upward from self-forcing, i.e. buoyancy, shear induced overturning

• Passive; No longer growing, residual cloud

Cumulus Clouds

• Shallow Cumulus (cumulus, scatted cumulus, strato-cumulus)– Depth small compared to scale height of troposphere, i.e.

– Usually confined to Planetary Boundary Layer (PBL)

– Typically non-precipitating

– Surface friction plays critical role to organization

• Deep Cumulus (congestus, cumulonimbi)– Depth comparable to scale height of troposphere

– Precipitating

– Friction plays secondary role to organization

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Instabilities Resulting in Cumulus

• Three basic atmospheric flow instabilities:1. Inertial Instability: Against horizontal inertial balance,

i.e. horizontal pressure gradient, coriolis and centrifugal force

2. Static Instability (absolute instability): Against vertical hydrostatic balance, i.e. vertical pressure gradient and gravity force

3. Symmetric Instability: Against inertial balance on an isentropic (constant potential temperature) surface, i.e. isentropic pressure gradient force, coriolis and isentropic centrifugal force (a combination of 1 and 2! Think about this!)

Symmetric Instability

Instabilities Resulting in Cumulus

• Conditional, i.e. only if saturated:– (CI) Conditional (static) instability – (CSI) Conditional Symmetric Instability

• Frictional, i.e. in the PBL:– Rayleigh- Bernard – Inflection Point Instability

• (KH) Kelvin-Helmholtz

– Gravity Wave Resonance

Kelvin-Helmholtz Instability

• Small perturbation tends to amplify by the advection of vorticity (shear => curvature)

• Resistance to growth of wave by static stability, i.e. Brunt-Vaisalla Frequency, N

• Condition for instability: 2 2 2

1

4i

ggN zzRu u u

z z z

Rayleigh-Benard Instability

• Results when a thin layer of fluid is subjected to heat fluxes from top or bottom of layer

• Forces:– Promoting overturning: heat flux– Resisting overturning: friction

Rayleigh Number

• Non-dimensional number depicting ratio of heat flux or buoyancy forcing to frictional resistance:

– h is fluid depth (m)– γ is the lapse rate (K/m)– αe is the coefficient of expansion– Dis the viscosity (m2/s)– K is the thermal conductivity (K m/s)

4e

a

h gR

DK

Condition forRayleigh-Benard Instability

• Linearize Navier stokes equations

• Assume wave solution:

• Then the condition for instability is:

' ' ( )

' ' ' ( )

, sin

, , cos

i kx ly t

i kx ly t

w T mze e

u v p mze e

32 2 2 4

2 2a

k l m hR

k l

Condition for Rayleigh-Benard Instability

• Instability for any number of combinations of k and l including:– Cells – Rolls

• The value that Ra must exceed is a function of horizontal wave number 2 2k l

0, 0k l

k l

Most Unstable Rayleigh-Benard Mode

• Differentiate stability condition w.r.t. horizontal wave number and set to zero to obtain condition for maximum (growth rate):

• If for simplicity we assume and we assume square cells where S is the spacing, then a ratio of horizontal spacing to depth S:h=3:1 is implied.

42

2 2 2 4 2 2 21 aRm m

k l m h k l m

2 /k l S /m h

Hexagonal form to Rayleigh Benard Convection

Organization of Boundary Layer Convection

• Cellular (Rayleigh-Benard)– Closed Cells

– Open Cells

• Linear– Wind Parallel (Rayleigh-Benard)

– Inflection Point (Kelvin-Helmholtz)

– Gravity Wave Resonance

• Spoked (Rayleigh-Benard)– Actinae

Cellular Convection

Mesoscale Cellular

Convection (MCC)

Linear Convection

Open, Closed

and Actinae Convect

ion

Linear,Roll-type

Convection

Hexagonal Cells

Cellular Convection

• Also known as mesoscale Cellular Convection (MCC)

• Two types:– Type I: typically to the east of continents during

the winter season over warm ocean currents (driven by heating from below)

– Type II: Occur during the summer to the west of continents over cool ocean currents (driven by cooling from above)

Cellular Convection

• Open vs. closed cells– Wintertime cold air masses that advect from continents

out over warm ocean currents produce convective marine PBLs. 

– Cold air masses that advect from continents out over warm ocean currents produce convective marine PBLs. 

– The convective cloudiness that evolves off-shore occurs as bands or streets and gives way downstream to chains  of open cells and then farther out to sea there are eventually  patterns of open and closed hexagonal convection

Cellular Convection

• Open vs. closed cells– This is the natural order to expect as unstable

convective PBLs are growing and near steady-state can develop in time with sufficient heating (and farther out to sea, which also finds the decreasing effects of vertical shear in the horizontal wind (<10-3s-1))         

Cellular Convection

• Actinae spoke-like cellular convection formations– Actinae do not occur in Type I CTBLs, because the

process  is too dynamic. 

– Actinae only occur in Type II CTBLs.  The reason being that in the Rayleigh-Prandtl regime stability  diagram there is a very small space (a narrow range of conditions that will support actinae). 

– In the Type II case the atmosphere is functioning in such a slow dynamic mode that the  actinae can be achieved.  It is easy to produce the actinae in the laboratory. 

Cellular Convection

• Actinae spoke-like cellular convection formations– The spoke-pattern convection is a geometric plan-form that

represents a transition between the open and  closed cellular convection patterns.  It is a transitional pattern and that is why it is always found between regions of  open and closed cells. 

– In thermal convection (both theory and  lab results) you can develop 6-arm patterns (linear mode for  weakly supercritical Rayleigh) to 12-arm patterns (non-linear mode). 

– Rotation would be no surprise because background vertical vorticity gets stretched  and the convective overturning (especially during transition from open to closed structure) produces horizontal vortex tubes  as well. 

Actinae