MODELLING VERTICAL STRUCTURE OF PHYSICAL FIELDS IN GULF OF RIGA

Juris Seņņikovs, Uldis Bethers, Laboratory for mathematical modeling of environmental and technological processes, University of Latvia, 8 Zeļļu, LV 1002, Rīga, ph. 7615711, e-mail bethers@latnet.lv

KEYWORDS: Upper mixed layer, stratification, atmosphere – ocean interface, mathematical modelling

ABSTRACT

The mathematical model for description of the time development of the vertical structure of temperature and salinity is developed. Model is based on the heat, mass and turbulent kinetic energy balance of the upper mixed layer; it includes description of the ice formation dynamics. Model accounts for the exchange with atmosphere, Baltic Proper, and for river run – off. The model is applied for simulation of the Gulf of Riga (1992 to 1995) comparing modelling results with observations.

INTRODUCTION

The model of the time-development of the vertical structure of the physical fields, i.e. temperature and salinity in the Gulf of Riga was developed by (Seņņikovs and Bethers, 1995). It is enhanced in the present paper adding the new features:

• ice formation and retreat with account for partial coverage of the Gulf of Riga by the ice and including different parameterisation of the fluxes on the sea-air and ice-air interfaces;
• limitation of the depth of the inflow from the Baltic Proper due the upper Baltic water is not dense enough to penetrate the deepest layers of the Gulf in summers;
• salt and heat carried by the riverine water;
• improved insolation parameterisation accounting for the sun height and cloudiness, thus enhancing the time resolution until ~1 hour.

The ice model inclusion modifies both the principal scheme of the physical model and its mathematical formulation as described in the following Chapters. The principal scheme of the accounted fluxes is given in Fig. 1. A numerical solution is performed by means of modified method (Реснянский, 1989).

PHYSICAL MODEL

Observations (Bērziņš, 1995) indicate presence of a distinct layer with a sharp gradient of the physical parameters in the Gulf of Riga from April till August. The depth profiles of the temperature and salinity include a mixed (quasi-homogenous) upper layer, a thin layer of the change of parameters (thermocline, halocline, pycnocline) and a lower stratified layer. The vertical differences in the physical fields are higher than horizontal, except for the hydrological fronts near rivers’ mouths. Contrary, from November till March the Gulf of Riga is almost vertically homogeneous. This, as well as presence of the ice cover almost every year characterises the difference between the shallow and brackish Gulf and the neighbouring Baltic Proper. The minor horizontal gradients in the main water mass of the Gulf allow a principal description of the seasonal cycle of the physical parameters by a one-dimensional model with the vertical resolution. The integral model of the upper layer similar to (Stigebrandt, 1985; Niiler and Kraus, 1977) is used in the present paper assuming the homogeneity of the physical parameters in this well-mixed layer. The turbulent heat and mass transfer equations are used to describe the physical fields below it.

The upper quasi-homogenous layer is intensively turbulent. Its presence and dynamics relate with the balance of the turbulent kinetic energy (TKE), which is generated mechanically, mainly by the wind and breaking wind waves. TKE is consumed for the mixing the incoming buoyancy flux (fresh water inflow, heat flux through the air-sea interface, ice melting of the ice, etc.), and for the entrainment of the denser and less turbulent water of the lower layer, if the upper layer deepens. The dynamic of the upper layer is affected also by the thermohaline convection in the case of a negative buoyancy flux (cooling above and heating below the minimum density temperature, ice freezing).

The balance of the water mass, heat and salt contents of the upper layer is constructed accounting for the fluxes that are either almost independent on the physical parameters of the Gulf or determined by these parameters. First group includes solar radiation, river run-off, outflow to the Baltic Proper. Inflow from the Baltic Proper has to be prescribed, too; however, its depth distribution is essentially dependent on the stratification of the Gulf. The heat and mass fluxes on the water-air and ice-air surfaces (evaporation/condensation, back radiation, sensible heat flux) depends not only on the external forcing parameters (humidity, wind, cloudiness, air temperature) but also on the temperature of the upper layer and the ice surface temperature. The ice usually covers some part of the Gulf during winters. The physical model under consideration assumes a constant temperature of the upper layer during the ice periods and different parameterisation of the heat fluxes for the ice-free and ice-covered parts of the Gulf. Mechanical generation of the TKE is considered only through the ice-free surface.

MATHEMATICAL MODEL

The balance equations are derived from the basic conservation laws. Mass balance of the upper layer neglecting precipitation/evaporation due their approximate equality is

(1)

here h is the depth of the upper mixed layer; w_e is the velocity of the entrainment of the water below the mixed layer into the latter; w(z) is the velocity of the vertical advection.

Heat balance for the upper layer

(2)

here r ₀ is the reference density (1000 kg·m^-3); c_p is the isobaric heat capacity of water (4200 J× m^-1K^-1); V(z) is the volume of the water above the depth z; A(z) is the cross section area at the depth z; T_S, T_F, T_in(z) are the temperatures of the upper layer, river water and the Baltic Proper water, respectively; D T is the difference between the temperature of the upper layer and the water just below it; F, Q_out are the mass fluxes of the rivers and flow to the Baltic Proper; q_in(z) is the depth distribution of the inflowing mass flux from the Baltic Proper. The terms on the right hand side of (2) denote, respectively (see also Fig. 1) (i) heat flux through the atmosphere-sea interface Q_ws including the insolation absorbed in the upper layer; (ii) heat flux through the ice-sea interface (if any) Q_iw; (iii) change of the heat content due to the mass fluxes inside the system; (iv) diffusive heat flux to the lower layer; (v) advective heat fluxes from the rivers and from /to the Baltic Proper. The temperature of the upper layer remains at the freezing temperature T_f (assumed constant T_f=-0.3°C) during the ice freezing and melting periods.

Salt balance for the upper layer

(3)

here Q_if is the heat flux for the melting of ice; M is the latent heat of the sea ice freezing, (M=3.32× 10⁵ J× kg^-1); S_S, S_ice, S_F, S_in are the salinities of the upper layer, sea ice, river and Baltic Proper water, respectively; D S is the difference between S_S and the salinity just below the upper layer; is the density of the diffusive salt flux to the lower layer.

The heat balance for the ice neglects the heat content of the ice cover

(4)

here Q_is is the heat flux on the atmosphere - ice interface, positive for the incoming flux.

The equation for the TKE balance stands as

(5)

here the left hand side is the energy that is necessary for the increasing of TKE
(e ² = 1× 10^-4 m²s^-2 ) and potential energy of the entraining water. This energy equals to zero during the retreat of pycnocline. D r is the difference between water density in upper layer and just below it. The terms on the right hand side of (5) describe, respectively:

1. The generation of the TKE, here 0£ c_i £ 1 is the part of the Gulf covered by the ice; empirical constant m_0» 3 (Seņņikovs and Bethers, 1995), u_* is the friction velocity (Gill, 1982)

(6)

here r _a is the air density (1.26 kg·m^-3); C_w=1.1× 10^-3 is the wind drag coefficient; W is the wind velocity.

1. The energy consumed for the mixing of the surface buoyancy flux (Q_B>0) or the TKE generated by the thermohaline convection (Q_B<0). The buoyancy flux is given by

3. The energy consumption U_in to mix the inflowing Baltic Proper water

(9)

here r _S=r (T_S,S_S), r _in=r (T_B, S_B) are the densities of the upper layer and the inflow.

4. The dissipation of the TKE due to convective motion

Turbulent energy generation by the shear stress on the interface between upper and lower layers is neglected as well as the radiation of the turbulent energy to the lower layer. The penetration of the mechanically generated turbulence is limited due to the Earth rotation by the thickness of the Ekman boundary layer h_Ek given (Stigebrandt, 1985) as

(11)

here H_k» 0.6 is an empirical constant (Seņņikovs and Bethers, 1995); f=1.22× 10^-4s^-1 is the Coriolis parameter at latitude 57° N. This limitation does not concern the turbulence generated by the convective motion.

The turbulent heat and mass transfer equations are used for calculating T(z) and S(z) for the lower layer z>h

(12)

here k _T, k _S are the turbulent heat and mass exchange coefficients, k _{T» k S} » 10^-6..10^-5 m²s^-1.

The boundary conditions for (12) in the absence of the exchange with the bottom are

(13)

here ' denotes ¶ /¶ z.

The heat and mass fluxes are supposed to be proportional to D T and D S, respectively

(14)

here empirical parameters are estimated as l_T» l_S» 1..100 m.

The heat fluxes Q_ws and Q_is through the sea-atmosphere and ice-atmosphere interfaces are formulated similarly

(15)

here I_w and I_i are the densities of the insolation flux absorbed in the water and ice, respectively. They are calculated using tables of insolation (Гидрометеоиздат, 1975). The solar radiation is assumed being absorbed in the upper mixed layer or the ice cover.

The back radiation from both surfaces is written as (Gill, 1982)

(16)

here s is Stephan-Boltzmann constant (5.67× 10^-8 W·m^-2K^-4); e_a is the partial pressure of the water vapour in the air (mbar); n_c is cloudiness; e _w=0.985, e _i=0.96 are emissivities of the water and ice, respectively; T stands either for the water or ice surface temperatures T_S or T_ice.

The latent heat fluxes (l =2.5× 10⁶ J× kg^-1 is the specific latent heat of vaporisation of the water, L=l +M=2.82× 10⁶ J× kg^-1 is the specific latent heat of ice sublimation) are calculated assuming evaporation parameterisation (Gill, 1982; Kraus, 1972)

(17)

here C_Ew» C_Ei» 1.1× 10^-3 are empirical water vapour exchange coefficients over the sea and ice, respectively; q_a is the absolute air humidity, q_aw, q_ai are the absolute humidities of the air-sea and air-ice interfaces. They are assumed to be equal to the saturation values of the water vapour at temperatures T_S or T_ice, respectively.

The sensible heat fluxes are written as (Gill, 1982; Kraus, 1972)

(18)

here c_pa = 1000 J× kg^-1K^-1 is isobaric heat capacity of air; C_Hw» C_Hi» 1.1× 10^-3 are empirical heat exchange coefficients over the sea and ice, respectively; T_a is the air temperature.

The dependence of the absolute humidity on the water vapour pressure must be added to (15-18)

(19)

here m » 0.62 is the ratio of the molar weights of water and air, p_a is the air pressure (assumed p_a=1000 mbar). The saturation pressure dependencies on the temperature are necessary for calculating the saturation values of the humidity. They are given for the water vapour above, respectively, the sea and ice surfaces as (Gill, 1982; Kraus, 1972)

(20)

here T is a expressed in ° C.

Assumption of the time-independent water level of the Gulf leads to a simple volume balance equation

(21)

The time-dependence of fluxes Q_in and Q_out can be found by an external model, (Bērziņš et. al, 2000). The outflow to the Baltic Proper is assumed from the upper layer, but an appropriate model is needed for q_in(z), the distribution of Q_in over the depth. Then the expression for w(z) follows from the continuity and incompressibility:

(22)

Let us assume (i) the depth independent temperature and salinity of the inflowing water T_in(z)=T_B, S_in(z)=S_B; (ii) the regular distribution by volume of the inflowing water between the sill depth h_sill and the equilibrium depth h_eq which can be found from

(23)

The density of the inflow then stands as

while the dependence of the vertical advection on z is

(25)

Accounting the additional turbulent energy consumed for the mixing of the inflowing water, (9) can be rewritten as

(26)

The ice conditions are described in the simple way, neglecting the variety of the ice forms and the snow cover over the ice. Then the ice conditions can be characterised by the area of the Gulf covered by the ice A_ice and the ice volume V_ice. Introducing the coefficient c_i of the ice coverage and the effective ice thickness as

(27)

one can derive the equation for the dynamics of the ice amount

(28)

Two additional hypotheses concerning the relationships between c_i, and the observed ice thickness h_ice are necessary for the closure of (28). The relationship

(29)

describes the ice volume distribution between the continuous ice cover and the drifting ice. The simplified assumption of no drifting ice

(30)

is adopted in this paper. The temperature of the ice surface, necessary for calculating mass and heat fluxes can be found assuming the linear temperature distribution across the ice cover. Obviously, this temperature cannot be greater than the freezing temperature

(31)

here k _ice» 2.1 J× K^-1m^-1s^-1 is the heat conductivity of the ice.

Another hypothesis concerns the dynamics of the ice-covered area of the Gulf. It can be found from the long-term observations of the ice thickness and coverage (Fig. 2). It can be deduced that for the most cases both freezing and melting of the ice proceeds with a linear relation between h_ice and c_i, however, the respective coefficients of the linear relationship are either constant (during freezing) or dependent on the ice distribution at the beginning of the melting period. The above considerations may be formalised as

(32)

where h_cov» 10 cm is the mean of the minimum value of the ice thickness for that the full coverage of Gulf is observed, see model behaviour of (32) in Fig. 2.

RESULTS AND DISCUSSSIONS

The developed model was applied for the real – time simulation of the vertical TS structure of the Gulf of Riga for time period May/92 to Oct/95. The following data were used for the forcing of the model of the dynamics of the vertical structure of the temperature and salinity:

1. Initial conditions, i.e. the temperature and salinity depth distributions from the sampling at 121 hydrological station on 19th May, 1992, at 18⁰⁰.
2. Meteorological conditions: cloudiness, air temperature and humidity, wind velocity from the observations at the Riga station, eight times daily.
3. The river run-off, inflow of the Baltic Proper water, and its salinity as (Bērziņš et al, 2000).

Data was linearly interpolated for their hourly values. The insolation was calculated from the astronomic considerations.

The calculated time development of the temperature depth distribution is shown in Fig. 3, whilst the time dependence of the surface and bottom temperature including comparison with the observations at station 121 is given in Fig. 4. The calculated development of fluxes on the atmosphere-sea interface is also shown in Fig. 4. Analysing calculations one can conclude that

• the forcing with high time-resolution reasonably reflects short-term meteorological peculiarities and single events, for instance, storms;

Fig.4. Time-dependence of calculated and observed surface (Ts) and bottom (Tb) temperatures (upper graph). Calculated daily means of heat flux (Q) components: insolation (I0), back radiation (Qbr), sensible (Qs) and latent (Ql) heat fluxes (lower graph).

• each of individual years has different scenario of stratification development;
• the calculation results indicate a reasonable agreement with the measurements at single station, all basic characteristics as the surface and bottom temperatures, the moments of spring retreat of pycnocline and full mixing of Gulf in autumns, the duration of the ice periods and their existence are present;
• the model is sensitive to the spring development of the pycnocline that determines the bottom temperatures during summers;
• the calculation results are suitable for the filling of gaps between the measurements, for instance to calculate the primary production;
• the components of the heat flux on the sea-air interface are significant secondary output from the model; the high time-resolution of them allows their utilization for different purposes, as regional climate models etc.

Model indicates a presence of two ice periods. First (winter 1992/93) is caused by the continuous increased cooling of the water masses during autumn-92, especially Oct-92 (see Fig. 4 for respective heat fluxes) but the second (winter 93/94) by the intensive cooling during Nov-93 and cold Feb-94. Relatively mild winter winds of 1994 were favourable for ice conditions due to the establishment of reverse stratification (see Fig. 3).

ACKNOWLEDGEMENT

This work has been supported by grant No. 96.0326 of Latvian Science Council. Authors owe V.Bērziņš (Latvian Fisheries Research Institute) for the useful discussions concerning the ice conditions over the Gulf of Riga, and access to the hydrometeorological and hydrological data series at LatFRI.

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