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River Ice Processes

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Advances in Water Resources Management

Part of the book series: Handbook of Environmental Engineering ((HEE,volume 16))

Abstract

The presence of ice in rivers is an important aspect to be considered in the development of water resources in cold regions. River ice research has largely been driven by engineering and environmental problems that concern society. Ice formation can affect the design, operation, and maintenance of hydraulic engineering facilities, in addition to issues related to ecological, environmental, and morphological aspects of the river. River ice processes involve complex interactions between hydrodynamic, mechanical, and thermal processes. They are also influenced by weather and hydrologic conditions. This chapter gives a brief overview of river ice processes, followed by discussions on the state of knowledge of these processes from freeze-up to breakup, and sediment transport in rivers under ice conditions.

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Abbreviations

A :

Flow cross sectional area under the cover, m2

A s :

Mass rate of snowfall over a unit area of water surface, \( \mathrm{kg}\;{\mathrm{m}}^{-2}\;{\mathrm{s}}^{-1} \)

a 0 :

Mean area of frazil crystal discs, mm

B :

Top width of channel, m

B a :

River surface width in the wind direction, m

B o :

Open water width, m

C :

Cloud cover in tenths

C a :

Area concentration of surface ice

C b :

Fraction of bed width covered by anchor ice

C i :

Specific heat of ice, \( 4.1855\times {10}^3\;\mathrm{J}\;{\mathrm{kg}}^{-1}\;{}^{\circ}\mathrm{C} \)

C p :

Specific heat of water, \( 4.215\times {10}^3\;\mathrm{J}\;{\mathrm{kg}}^{-1}\;{}^{\circ}\mathrm{C} \) check p. 25?

C v :

Suspended ice concentration

C g v :

Suspended ice concentration due to thermal growth

C o :

A reference sediment concentration at a near bed reference level z o , cm

C w :

Wind drag coefficient on water surface

C z :

Chezy’s coefficient, \( {\mathrm{m}}^{0.5}\;{\mathrm{s}}^{-1} \)

C a,max :

Maximum allowable surface ice concentration

C s :

Suspended sediment concentration, mg/L

\( {c}_f=\frac{n_b^2g}{{\left({\alpha}_bH\hbox{'}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} \) :

Bed friction coefficient

\( {c}_{iw}=\frac{n_i^2g}{{\left({\alpha}_iH\hbox{'}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}} \) :

Coefficient of water drag on ice

D :

Sediment particle diameter, cm

D H :

Hydraulic diameter, m

\( {D}_{*}={D}_{50}{\left[\left(s-1\right)g/{\nu}^2\right]}^{1/3} \) :

A dimensionless sediment particle parameter

D 50 :

Median diameter of sediment, cm

d e :

Frazil crystal thickness, mm

\( {d}_n \) :

Nominal diameter of under-cover granular ice, cm

d a :

Day number of the year counting from January 1

E :

Net volumetric rate of loss of frazil due to exchange with the surface layer and the anchor ice, m3/s

E a :

Rate of change of surface ice concentration due to sources and sinks, s−1

E m :

Rate of change of surface ice mass per unit area of water surface due to sources and sinks, kg/m

E o :

Eccentricity correction factor of the earth’s orbit

e a :

Vapor pressure, mb

e an :

Porosity of anchor ice

e c :

Porosity in the accumulation between ice floes

e f :

Porosity of the frazil ice portion of surface ice floes

\( {e}_j={e}_c+\left(1-{e}_c\right){e}_p \) :

Overall porosity of the surface ice accumulation

e p :

Porosity of ice floes

e s :

Saturated vapor pressure above the river surface, mb

e so :

Saturated vapor pressure corresponding to the river surface temperature, mb

\( {e}_T={C}_p\rho \left(1-{C}_v\right){T}_w-{\rho}_i{C}_v{L}_i \) :

Thermal energy of the ice–water mixture per unit volume, J/m3

e u :

Porosity of the frazil jam

F :

Shape factor of under-cover granular ice

\( {\overrightarrow{F}}_a \) :

Wind drag on ice, N/m2

F b :

Inter-bed particle resistance per unit area, N/m2

F rc :

A critical Froude number beyond which juxtaposition of ice floes cannot occur

F rp :

A limiting Froude number for cover progression

\( {\overrightarrow{F}}_w \) :

Water drag on ice, N/m2

f :

Fraction of the width covered by border ice cover

\( \overrightarrow{G} \) :

Gravitational force along the water surface slope, N/m2

g :

Gravity, m/s2

H :

Water level, m

H T  = H R  + η :

Total water depth from channel bottom to the water surface, m

H′ = H R  + η′:

Water depth beneath the ice layer, m

H R :

Water depth below the reference level, m

H t :

Water depth underneath an equivalent ice–water interface, m

\( {h}_e={h}_i+{h}_f\left(1-{e}_f\right) \) :

Equivalent thickness of surface ice floes, m

h f :

Thickness of the frazil accumulation on the underside of surface ice floes, m

h i :

Thickness of the solid portion of surface ice floes, m

h i,p :

Thickness of the porous portion of the ice cover, m

h iws :

Heat exchange coefficient between anchor ice and the substrate flow, W/m2 °C

h s :

Snow cover thickness on top of the ice cover, m

h sa :

Heat transfer coefficient at the snow–air interface, W/m2 °C

h sb :

Substrate flow depth at the bottom of the anchor ice, m

h wi :

Heat transfer coefficient between water and ice cover, W/m2 °C

I so :

Solar constant, 1367 W/m2

K w :

Thermal conductivity of water, W m−1 °C−1

k i :

Thermal conductivity of ice cover, W m−1 °C−1

k ic :

Ice cover roughness height, m

k n :

Decay constant of the ice cover roughness coefficient, day−1

k s :

Thermal conductivity of snow cover, W m−1 °C−1

L i :

Latent heat of fusion of ice, \( 3.3484\times {10}^5\;\mathrm{J}\;{\mathrm{kg}}^{-1} \)

\( {M}_i={\rho}_i{C}_a{t}_i \) :

Surface ice mass per unit area, kg/m2

m :

Optical air mass

m o :

Optical air mass at sea level

N f :

Number of frazil particles per unit volume, 1/m3

\( {N}_u={h}_{wi}{D}_H/{K}_w \) :

Nusselt number

n b , n i , and n c :

Bed, ice cover, and composite Manning’s coefficients, respectively, s ft−1/3

n io :

Single layer surface ice roughness, s ft−1/3

n o , n e :

Initial and end values of the ice cover roughness coefficient, s ft−1/3

P :

Internal pressure of pack ice, Pa

\( {P}_r={C}_p\mu {K}_w \) :

Prandtl number

p a :

Atmospheric pressure, mb

p b and p i :

Wetted perimeters formed by the channel bed and the ice cover, respectively, m

p o :

Atmospheric pressure at sea level, mb

Q :

Water discharge, m3/s

Q i d , Q i s :

Volumetric rates of suspended and surface ice discharge, respectively, m3/s

Q u :

Volumetric rate of ice entertainment under the cover at the leading edge, m3/s

q f :

Heat transfer between water and a frazil particle per unit frazil surface area, J/cm2 s

q b :

Volumetric rate of bed sediment transport per unit width, m3/s

q c :

Volumetric rate of undercover ice transport per unit width, m3/s

\( {\overset{\rightharpoonup }{q}}_l \), and \( {\overset{\rightharpoonup }{q}}_u \) :

Unit-width water discharge beneath the ice layer, and in the ice layer, respectively, m2/s

q s :

Seepage flow in the surface ice layer, m3/s

\( {\overset{\rightharpoonup }{q}}_{ice} \) :

Unit-width surface ice discharge, m3/s

R :

Hydraulic radius, m

R a :

Rate of change of surface ice concentration due to mechanical redistribution of the ice mass, 1/s

R an :

Contribution to the surface ice run from anchor ice release, m3/s

R b :

Rate of lateral growth of border ice, m/h

R c :

Radius of curvature of the centreline of the ice sheet, m

R i :

Hydraulic radius associated with the ice cover, m

\( {R}_{eH}=\rho U{D}_H/\mu \) :

Reynolds number

R t :

Albedo of river surface

\( \overrightarrow{R} \) :

Internal ice resistance due to floe-to-floe interactions, N/m2

RH :

Relative humidity in percentage, %

r s :

Reflectivity of long wave radiation from the river surface

S f :

Friction slope of the channel

S w :

Water surface slope

S 5 :

Cumulative degree-days above −5 °C, °C day

\( s={\rho}_s/\rho \) :

The ratio of sediment density and water density

T :

Temperature in the cover, °C

T a :

Air temperature, °C

T cr :

Supercooled water surface temperature below which static skim ice will form, °C

T ak :

Air temperature at 2 m above the surface, in the absolute scale, K

T m :

Freezing point of water, 0 °C

T s :

River surface temperature, °C

T t :

A sediment transport stage parameter

T sk :

River surface temperature in the absolute scale, K

T sn :

Snow surface temperature, °C

T w :

Cross section-averaged water temperature, °C

T w,s :

Water surface temperature, °C

t eq :

Equilibrium ice jam thickness, defined as the jam thickness in the uniform reach, m

t f :

Thickness of frazil jam, m

t i :

Surface ice layer or ice cover thickness, m

t′ i :

Submerged surface ice layer thickness, m

u * i :

Shear velocity on the undersurface of the frazil jam, m/s

v a2 :

Wind velocity at 2 m above the surface, m/s

v b :

Buoyancy velocity of frazil ice, m/s

v 0 :

Mean volume of frazil crystals, cm3

v ' z :

Vertical component of turbulent fluctuation velocity, m/s

U :

Cross-section-averaged flow velocity, m/s

U s :

Width-averaged surface ice velocity, m/s

V a :

Wind velocity, m/s

V cb :

Maximum velocity at which a surface ice floe can adhere to the border ice edge, m/s

V cp :

Rate of progression of leading edge, m/s

\( \widehat{V} \) :

Visibility, km

\( \forall \) :

Volume of anchor ice per unit bed area, m

\( {\overset{\rightharpoonup }{V}}_i \) :

Surface ice velocity, m/s

\( {\overset{\rightharpoonup }{V}}_w \) :

Water current velocity underneath the ice cover, m/s

W b :

Submerged weight of bed materials per unit area, N/m2

W i :

Width of ice sheet detached from banks, i.e. the distance between hinge cracks, m

x, y, and t :

Space and time variables, m and s

α :

Probability of deposition of frazil particles reaching the surface layer

α b :

Fraction of the water depth affected by the bed friction

α i :

Fraction of the water depth affected by the ice friction

α z :

Solar latitude, angular elevation of the sun above the true horizon, degrees

β :

Coefficient quantifying the rate of re-entrainment of surface ice per unit area, 1/s

γ :

Coefficient quantifying the rate of accretion to the bed per unit area, m/s

\( {\gamma}_e=0.5{\rho}_ig\left(1-{\scriptscriptstyle \frac{\rho_i}{\rho }}\right)\left(1-{e}_j\right) \) :

N/m3

ΔB :

Growth of border ice width for a given time step, m

Δ d :

Dune height, m

Δ i :

\( \mathrm{The}\ \mathrm{ratio}\left(\rho -{\rho}_i\right)/\rho \)

Δϕ*:

Surface heat exchange during a time step, \( \mathrm{J}\;{\mathrm{m}}^{-2} \)

δ :

Solar declination, in radians

ε a :

Emissivity of atmosphere

ε s :

Emissivity of river surface

ξ i :

Bulk extinction coefficient, 1/cm

η :

Water surface elevation, m

η′:

Elevation of the bottom of surface ice, m

\( \tilde{\eta}=\frac{\varTheta }{\varTheta_c} \) :

Ratio of θ to the critical value of θ for incipient motion

\( \varTheta =\frac{\tau_b}{\Delta \rho gD} \) :

Dimensionless flow strength for sediment transport

Θ c :

Critical value of Θ for incipient motion

Θ i :

Dimensionless flow strength for cover load transport

Θ ci :

Critical flow strength for undercover ice transport

θ a :

Angle between the wind direction and the x-axis, degree

λ s :

Seepage coefficient in the pack ice, m/s

μ :

Dynamic viscosity of water, N s/m2

μ i :

Ice-over-ice friction coefficient

\( \widehat{\nu} \) :

Nonlinear shear viscosity of the surface ice run

п:

A stress parameter for ice cover during breakup period, kPa

ρ, ρ a , ρ i , and ρ s :

Density of water, air, ice, and sediment, respectively, kg/m3

σ:

Stefan–Boltzmann constant, 5.67 × 10−8 W m−2 K−4

σ f :

Flexural strength of the ice cover, kPa

σ ij :

Internal ice stress in the surface ice layer, N/m2

τ a :

Wind drag on the water or ice surface along the channel, N/m2

τ ai :

Wind drag on ice, N/m2

τ aw :

Wind drag on water, N/m2

τ b , τ i :

Shear stresses at the channel bottom and the ice–water interface, respectively, N/m2

τ s :

Shear stresses at the ice–water interface, N/m2

\( {\tau}^{\left(a-w\right)} \) :

Wind shear stress on water surface, N/m2

\( {\tau}^{\left(i-w\right)} \) :

Shear stress at the ice–water interface, N/m2

τ c :

Cohesion in the shear stress of floating surface ice accumulation, N/m2

τ g :

Weight component of the ice cover along the surface slope, N/m

τ w :

Water drag, N/m2

\( {\tilde{\tau}}_i \) :

Driving stress on the ice cover during breakup period, N/m2

Φ i :

Dimensionless under-cover ice transport capacity

\( {\varPhi}_b=\frac{q_b}{\sqrt{\Delta g{D}^3}} \) :

Dimensionless bed load intensity

φ :

Internal friction angle of surface ice, degree

ϕ b :

Bed heat flux per unit area, W/m2

ϕ ba :

Atmospheric long-wave radiation reaching the river surface, kJ/m2 h

ϕ br :

Reflected long wave radiation from the river surface, kJ/m2 h

ϕ bs :

Long wave radiation emitted by the water surface, kJ/m2 h

ϕ B :

Effective back radiation or terrestrial radiation, kJ/m2 h

ϕ cl :

Incoming short wave radiation under clear skies, kJ/m2 h

ϕ E :

Evaporation heat transfer, kJ/m2 h

ϕ H :

Sensible heat transfer, kJ/m2 h

ϕ ia :

Heat loss at the air-ice interface, kJ/m2 h

ϕ P :

Heat transfer due to precipitation on the water surface, kJ/m2 h

ϕ ps , ϕ pz :

Solar radiation on the ice cover surface, and at a depth below the ice surface, kJ/m2 h

ϕ ri :

Incoming solar radiation, kJ/m2 h

ϕ rr :

Solar radiation reflected back to the atmosphere, kJ/m2 h

ϕ R :

Net solar radiation, kJ/m2 h

ϕ si :

Net rate of heat loss from top and bottom of surface ice floes, kJ/m2 h

ϕ sk :

Rate of heat loss through top and bottom of the suspended ice layer, kJ/m2 h

ϕ so :

Total extraterrestrial solar radiation incident on a horizontal surface, kJ/m2 h

ϕ ss :

Rate of heat gain through top and bottom of the suspended layer, kJ/m2 h

ϕ v :

Rate of internal heating of the cover due to absorption of penetrated short wave radiation, kJ/m2 h

ϕ wi :

Heat flux between river water and ice cover, kJ/m2 h

ϕ*:

Total surface heat flux between air and water, ϕ wa , or between ice and air, ϕ ia , kJ/m2 h

ϕ * s :

Rate of heat loss at the snow surface, kJ/m2 h

ψ :

Latitude in degrees, north positive, south negative

ω :

Hour angle, degrees

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  1. Department of Civil and Environmental Engineering, Wallace H. Coulter School of Engineering, 232 Rowley Laboratories, Clarkson University, 8 Clarkson Avenue, Potsdam, NY, 13699-5710, USA

    Hung Tao Shen Ph.D.

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    Chih Ted Yang

  3. Lenox Institute of Technology, Newtonville, New York, USA

    Mu-Hao S. Wang

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Shen, H.T. (2016). River Ice Processes. In: Wang, L., Yang, C., Wang, MH. (eds) Advances in Water Resources Management. Handbook of Environmental Engineering, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-22924-9_9

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