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2.3 Stability & Convection

Atmospheric stability determines whether air parcels rise or sink. Unstable conditions lead to convection, clouds, and thunderstorms; stable conditions suppress vertical motion.

Static Stability

Absolutely Stable

Environmental lapse rate < moist adiabatic rate

Γ < Γ_m ≈ 6°C/km: Displaced parcel returns to original position

Absolutely Unstable

Environmental lapse rate > dry adiabatic rate

Γ > Γ_d = 9.8°C/km: Parcel accelerates away from original position

Conditionally Unstable

Γ_m < Γ < Γ_d

Most common state: stable for dry air, unstable for saturated air

Stability Criterion in Terms of Potential Temperature

$$\frac{\partial \theta}{\partial z} > 0 \;\;\text{(stable)}, \qquad \frac{\partial \theta}{\partial z} = 0 \;\;\text{(neutral)}, \qquad \frac{\partial \theta}{\partial z} < 0 \;\;\text{(unstable)}$$

Potential temperature increasing with height means the atmosphere is stably stratified.

Buoyancy & Brunt-Väisälä Frequency

Parcel Buoyancy

$$B = g\,\frac{T_{v,\text{parcel}} - T_{v,\text{env}}}{T_{v,\text{env}}}$$

Net upward acceleration on a displaced parcel, using virtual temperature $T_v$ to account for moisture effects. Positive $B$ = parcel accelerates upward.

Brunt-Väisälä (Buoyancy) Frequency

$$N^2 = \frac{g}{\theta}\frac{d\theta}{dz} = \frac{g}{T}\left(\frac{dT}{dz} + \Gamma_d\right)$$

Frequency of oscillation of a displaced parcel in a stably stratified atmosphere.

$N^2 > 0$

Stable: oscillation period $\tau = 2\pi/N$

$N^2 = 0$

Neutral stability

$N^2 < 0$

Unstable: exponential growth

Richardson Number

$$Ri = \frac{N^2}{\left(\dfrac{\partial u}{\partial z}\right)^2 + \left(\dfrac{\partial v}{\partial z}\right)^2} = \frac{N^2}{|\nabla_z \mathbf{u}|^2}$$

Ratio of buoyancy suppression to wind shear production of turbulence. When $Ri < 0.25$, shear instability (Kelvin-Helmholtz) can develop even in stable air.

CAPE & CIN

CAPE (Convective Available Potential Energy)

$$CAPE = \int_{LFC}^{EL} g \frac{T_{parcel} - T_{env}}{T_{env}} dz$$

Positive buoyancy: energy available for convection (J/kg)

• <1000: Weak instability
• 1000-2500: Moderate
• >2500: Strong (severe weather)

CIN (Convective Inhibition)

$$CIN = \int_{SFC}^{LFC} g \frac{T_{parcel} - T_{env}}{T_{env}} dz$$

Negative buoyancy: energy barrier to convection (J/kg)

• <50: Weak cap
• 50-200: Moderate cap
• >200: Strong cap (inhibits storms)

Maximum Updraft Speed from CAPE

$$w_{\max} = \sqrt{2 \cdot \text{CAPE}}$$

Theoretical maximum vertical velocity (m/s). For CAPE = 2500 J/kg, $w_{\max} \approx 71$ m/s. In practice, entrainment and drag reduce this substantially.

Level of Free Convection (LFC)

The altitude where a lifted parcel first becomes warmer than its environment. Formally defined where buoyancy changes sign:

$$T_{v,\text{parcel}}(z_{\text{LFC}}) = T_{v,\text{env}}(z_{\text{LFC}}) \quad \text{with} \quad \frac{\partial B}{\partial z}\bigg|_{\text{LFC}} > 0$$

Below the LFC, the parcel must be forced upward (e.g., by fronts, orography, or convergence). Above the LFC, positive buoyancy drives free convection up to the Equilibrium Level (EL).

Interactive Simulation: CAPE and CIN Computation

Python

Creates a synthetic atmospheric sounding and computes a lifted parcel ascent. Identifies LCL, LFC, and EL levels, numerically integrates CAPE and CIN, and displays a Skew-T style diagram with shaded buoyancy regions.

cape_cin_sounding.py103 lines

import numpy as np
import matplotlib.pyplot as plt

g = 9.81; R_d = 287.0; c_p = 1005.0; L_v = 2.5e6; R_v = 461.5; eps = R_d / R_v
Gamma_d = g / c_p  # K/m

def es(T): return 611.2 * np.exp(17.67 * (T - 273.15) / (T - 29.65))
def sat_mr(T, p): e = es(T); return eps * e / (p - e)
def moist_lapse(T, p):
    rs = sat_mr(T, p)
    return Gamma_d * (1 + L_v*rs/(R_d*T)) / (1 + L_v**2*rs/(c_p*R_v*T**2))

# Environmental sounding (conditionally unstable)
z = np.arange(0, 14001, 50)  # m
p_env = 101325 * np.exp(-z / 8500)
T_env = np.zeros_like(z, dtype=float)
T_env[0] = 303.15  # 30 C surface
for i in range(1, len(z)):
    if z[i] < 1500: lr = 9.0   # steep low-level lapse
    elif z[i] < 3000: lr = 5.0  # stable layer (cap)
    elif z[i] < 10000: lr = 7.5 # conditionally unstable
    else: lr = 4.0               # upper stable
    T_env[i] = T_env[i-1] - lr * 0.05 / 1000 * 1000  # per 50m step
    T_env[i] = T_env[i-1] - lr * 50 / 1000

# Parcel ascent from surface with Td = 22C
Td_sfc = 295.15
T_parcel = np.zeros_like(z, dtype=float)
T_parcel[0] = T_env[0]
saturated = False; lcl_z = None; lfc_z = None; el_z = None

for i in range(1, len(z)):
    dz = z[i] - z[i-1]
    if not saturated:
        T_parcel[i] = T_parcel[i-1] - Gamma_d * dz
        Td_est = Td_sfc - 1.8 * z[i] / 1000
        if T_parcel[i] <= Td_est:
            saturated = True; lcl_z = z[i]
    else:
        ml = moist_lapse(T_parcel[i-1], p_env[i-1])
        T_parcel[i] = T_parcel[i-1] - ml * dz
    if saturated and T_parcel[i] > T_env[i] and lfc_z is None and z[i] > (lcl_z or 0):
        lfc_z = z[i]
    if lfc_z and T_parcel[i] < T_env[i] and el_z is None and z[i] > lfc_z + 500:
        el_z = z[i]

# Compute CAPE and CIN
cape = 0; cin = 0
for i in range(1, len(z)):
    B = g * (T_parcel[i] - T_env[i]) / T_env[i]
    dz = z[i] - z[i-1]
    if lfc_z and z[i] >= lfc_z and (el_z is None or z[i] <= el_z) and B > 0:
        cape += B * dz
    elif z[i] <= (lfc_z or 1e10) and B < 0:
        cin += abs(B) * dz

fig, ax = plt.subplots(figsize=(10, 9))
fig.patch.set_facecolor('#0f172a'); ax.set_facecolor('#1e293b')
ax.tick_params(colors='#cbd5e1'); ax.xaxis.label.set_color('#cbd5e1')
ax.yaxis.label.set_color('#cbd5e1'); ax.title.set_color('#cbd5e1')
ax.grid(True, color='#334155', alpha=0.4)
for sp in ax.spines.values(): sp.set_color('#334155')

ax.plot(T_env - 273.15, z/1000, color='#ef4444', lw=2.5, label='Environment T')
ax.plot(T_parcel - 273.15, z/1000, color='#38bdf8', lw=2.5, label='Parcel T')

# Shade CAPE and CIN
for i in range(1, len(z)):
    zk = z[i]/1000; zk1 = z[i-1]/1000
    if lfc_z and z[i] >= lfc_z and (el_z is None or z[i] <= el_z) and T_parcel[i] > T_env[i]:
        ax.fill_betweenx([zk1, zk], [T_env[i-1]-273.15, T_env[i]-273.15],
                         [T_parcel[i-1]-273.15, T_parcel[i]-273.15], color='#ef4444', alpha=0.25)
    elif z[i] <= (lfc_z or 1e10) and T_parcel[i] < T_env[i]:
        ax.fill_betweenx([zk1, zk], [T_parcel[i-1]-273.15, T_parcel[i]-273.15],
                         [T_env[i-1]-273.15, T_env[i]-273.15], color='#3b82f6', alpha=0.25)

levels = [(lcl_z,'LCL','#4ade80','--'),(lfc_z,'LFC','#fbbf24','--'),(el_z,'EL','#c084fc','--')]
for lz, lbl, col, ls in levels:
    if lz:
        ax.axhline(lz/1000, color=col, ls=ls, lw=1.5)
        ax.text(-55, lz/1000+0.15, f'{lbl}: {lz/1000:.1f} km', color=col, fontsize=10, fontweight='bold')

ax.text(-15, 8, f'CAPE = {cape:.0f} J/kg', color='#fca5a5', fontsize=12, fontweight='bold',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='#1e293b', edgecolor='#ef4444'))
ax.text(-15, 7, f'CIN = {cin:.0f} J/kg', color='#93c5fd', fontsize=12, fontweight='bold',
        bbox=dict(boxstyle='round,pad=0.4', facecolor='#1e293b', edgecolor='#3b82f6'))

ax.set_xlabel('Temperature (\u00b0C)', fontsize=12); ax.set_ylabel('Height (km)', fontsize=12)
ax.set_title('CAPE & CIN: Skew-T Style Parcel Diagram', fontsize=14, fontweight='bold')
ax.legend(loc='upper right', facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax.set_xlim(-60, 35); ax.set_ylim(0, 14)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor=fig.get_facecolor())
w_max = np.sqrt(2 * cape) if cape > 0 else 0
print("=== CAPE and CIN Computation ===")
print(f"  Surface T = 30.0 \u00b0C, Td = 22.0 \u00b0C")
print(f"  LCL = {lcl_z/1000:.2f} km" if lcl_z else "  LCL = N/A")
print(f"  LFC = {lfc_z/1000:.2f} km" if lfc_z else "  LFC = N/A")
print(f"  EL  = {el_z/1000:.2f} km" if el_z else "  EL  = N/A")
print(f"  CAPE = {cape:.0f} J/kg")
print(f"  CIN  = {cin:.0f} J/kg")
print(f"  Max updraft w_max = sqrt(2*CAPE) = {w_max:.1f} m/s")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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